Extremal problems on graphs and codes (Ido Ben-Eliezer, Ph ... · Blavatnik School of Computer Science Extremal Problems on Graphs and Codes ... and ideas that continuously convinced

The Raymond and Beverly Sackler

Faculty of Exact Sciences

Blavatnik School of Computer Science

Extremal Problems

on

Graphs and Codes

Thesis submitted for the degree “Doctor of Philosophy”

by

Ido Ben-Eliezer

Under the supervision of

Prof. Noga Alon and Prof. Michael Krivelevich

Submitted to the Senate of Tel Aviv University

July 2011

Acknowledgments

One of the greatest benefits of working on this thesis was definitely the unique

opportunity to interact with many intelligent, bright and friendly people.

I would like to thank my advisors, Noga Alon and Michael Krivelevich for

their support and care. Our meetings were always valuable and enjoyable.

I started to work with Michael almost six years ago. Since then, his door

was always open, and I have tried to learn as much as I can from his great

problem solving skills and his advices in math and beyond.

Noga gave me an endless number of interesting (and difficult) problems

and ideas that continuously convinced me that choosing Combinatorics as

my research area was the right decision.

It was great to collaborate with Benny Sudakov, Simon Litsyn, Shachar

Lovett, Rani Hod and Ariel Yadin. I would like to thank them for many

hours of fruitful meetings which led to substantial parts of this work.

Such a long period would not be enjoyable without so many discussions

about everything (including even math) with my fellows Sonny Ben-Shimon,

Simi Haber, Rani Hod, Uri Stav, Danny Vilenchik and Amit Weinstein.

Finally, I would like to thank my family for their love and support during

the last four (or twenty seven) years; This part will not be complete without

thanking Inbal for making the last two years the best two.

Abstract

We study extremal problems related to codes, polynomials and graphs. The

problems we study have applications in theoretical computer science, Ramsey

theory and probabilistic combinatorics.

Part I concerns error correcting codes and polynomials, both over finite

fields and over the reals. Many constructions of families of error correct-

ing codes are formed by evaluating polynomials over finite fields on some

set of points. In particular, the set of degree-d polynomials forms the well

known Reed-Muller code (see, e.g., [93]). Some of our results can therefore

be interpreted both in terms of polynomials and in terms of error correcting

codes.

In Chapter 1 we study the minimal size of the support of a distribution Dof vectors over Fnp , such that for every polynomial Q over Fnp , the distribution

of Q(x) : x ∼ D is close to the distribution of Q(x) : x ∈ Fnp. We

provide nearly tight bounds on the minimum possible support size of such

a distribution. A key step in our proof is the construction of a large linear

space of degree-d polynomials with small bias.

In Chapter 2, we provide a new tail bound on the bias of a random

degree-d polynomial (equivalently, on the weight distribution of Reed-Muller

codes). Our results imply that a typical degree-d polynomial cannot be

approximated by polynomials of smaller degree. Moreover, our results can

be used to construct a very large linear space of polynomials with small bias,

which leads to a slight improvement of the results from Chapter 1.

Chapter 3 deals with threshold (or sign) functions of degree-d real poly-

nomials over the cube. We show, for every such function, the existence of a

small sub-assignment with no influential variables. Moreover, we prove that

vi Abstract

this implies that some families of functions cannot be approximated by low

degree polynomial threshold functions.

Part II deals with extremal and probabilistic problems in graphs. In

Chapter 4 we show tight results on the resilience of a (pseudo)random di-

rected graph with respect to the property of having a long directed cycle. In

particular, we show that even if one removes maliciously almost half of the

edges of such a graph, the obtained graph still has a directed cycle of linear

size.

Chapter 5 deals with the size Ramsey number of a directed path. Namely,

we prove nearly tight bounds on the minimal number m such that there is

an oriented graph G with m edges, with the property that every q-coloring

of E(G) contains a long directed cycle.

In Chapter 6 we study biased orientation games, in which two players,

Maker and Breaker take turns by directing previously undirected edges of Kn.

In our setting, at each round Maker orients one edge and Breaker orients at

most b edges. By the end of the game, all the edges are directed and hence

we have a tournament T . Maker wins if T has some predetermined property,

and Breaker wins otherwise. We study the bias b that guarantees Maker’s win

(or Breaker’s win) for the properties of containing a directed cycle, containing

a directed Hamilton cycle, and containing a fixed graph H.

In Chapter 7, we study the following Ramsey-type problem, motivated

by an extremal problem suggested by Wigderson. Given a fixed graph H,

what is the minimal number of colors k such that there are n colorings of

E(Kn) with k colors, each is associated with one of the vertices, such that

the following holds. For every copy T of H, one of the vertices of T defines

a rainbow coloring of E(T ). We characterize the set of graphs for which a

constant number of colors is sufficient. Moreover, we give a general upper

bound on the number of required colors for every graph H, and give some

lower bounds on certain graphs of special interest.

Finally, Chapter 8 deals with partitions of graphs G = (V,E) into two

sets V = V1∪V2 such that the two graphs G[V1], G[V2] have exactly the same

number of vertices and edges. In particular, we show that for every even n

and for every choice of p, a random graph distributed according to G(n, p)

has this property with high probability.

Contents

Acknowledgments iii

Abstract v

Introduction 1

I Codes and Polynomials 11

1 Small sample spaces cannot fool low degree polynomials 13

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.1 Lower bound . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.2 Upper bound . . . . . . . . . . . . . . . . . . . . . . . 20

1.3 A complex variant of Lemma 1.5 . . . . . . . . . . . . . . . . 21

1.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 22

2 Random low degree polynomials are hard to approximate 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.2 Our results . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . 30

2.2.1 Proof of Lemma 2.2 . . . . . . . . . . . . . . . . . . . . 31

2.2.2 Proofs of technical claims . . . . . . . . . . . . . . . . 33

viii CONTENTS

2.2.3 Proof of Lemma 2.4 . . . . . . . . . . . . . . . . . . . . 36

2.3 Proof of Proposition 2.3 . . . . . . . . . . . . . . . . . . . . . 39

3 Polynomial threshold functions: Structure and approxima-

tion 41

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.1 Decision trees . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.2 Tail estimates for polynomials . . . . . . . . . . . . . . 45

3.3 The effect of partial assignments . . . . . . . . . . . . . . . . . 46

3.4 Proof of Lemma 3.7 . . . . . . . . . . . . . . . . . . . . . . . . 48

II Graphs 55

4 Long cycles in subgraphs of (pseudo)random directed graphs 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.1.1 The models . . . . . . . . . . . . . . . . . . . . . . . . 60

4.1.2 Our results . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 The regularity lemma and long paths in regular pairs . . . . . 64

4.2.1 The regularity lemma . . . . . . . . . . . . . . . . . . . 64

4.2.2 Every regular pair contains a long path . . . . . . . . . 66

4.3 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 72

5 The size Ramsey number of a directed path 75

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3.1 Sparse graphs have large acyclic sets . . . . . . . . . . 80

5.3.2 Acyclic colorings and coloring acyclic sets . . . . . . . 81

5.4 Upper Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

CONTENTS ix

5.4.1 Pseudorandom digraphs . . . . . . . . . . . . . . . . . 88

5.4.2 The case of two colors . . . . . . . . . . . . . . . . . . 90

5.4.3 The general case . . . . . . . . . . . . . . . . . . . . . 93

5.5 Concluding remarks and open problems . . . . . . . . . . . . . 94

6 Biased orientation games 95

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3 The cycle game . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.4 The Hamiltonicity game . . . . . . . . . . . . . . . . . . . . . 101

6.5 The H-creation game . . . . . . . . . . . . . . . . . . . . . . . 109

7 Local Rainbow Colorings 115

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.3 Graphs requiring a bounded number of colors . . . . . . . . . 119

7.4 Upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.5 Lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.6 Concluding remarks and open questions . . . . . . . . . . . . 127

8 Perfectly balanced partitions of smoothed graphs 129

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.2 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Bibliography 137

Introduction

In this work we study some extremal and probabilistic aspects of error cor-

recting codes, polynomials and graphs. The problems we consider range from

real polynomials to Ramsey problems on graphs.

A key theme in this work is the frequent use of various probabilistic meth-

ods . This is rather natural when studying the behavior of random structures

(for example, random degree-d polynomials in Chapter 2 and random graphs

in Chapter 4 and Chapter 8). However, probabilistic methods are also used

in the study of deterministic objects. Namely, in order to show the exis-

tence of an object with some desired properties, one can define a distribution

and show that a randomly chosen object from this distribution has these

properties with positive probability. This approach is widely used in modern

combinatorics (see [13]). Here we follow this line with constructions of large

spaces of polynomials with small bias (Chapter 1 and Chapter 2), construc-

tions of objects with Ramsey-type properties (Chapter 5 and Chapter 7) and

design of strategies in orientation games (Chapter 6).

In what follows we provide a summary of the results that are presented

in this work.

Codes and Polynomials

Part I deals with some probabilistic and extremal aspects of polynomials and

error correcting codes. An error correcting code (over Fnp ) is a set of vectors

(or codewords) from Fnp . The minimal distance is the minimum among all

pairwise Hamming distances of codewords. For many practical and theoreti-

cal applications, codes with high rate (i.e, consisting of many codewords) and

2 Introduction

with large minimum distance are required. It is still a major open question

to determine the best possible rate for a given minimum distance (or vice

versa).

There are many algebraic constructions of error correcting codes that

use the special symmetric properties of polynomials over finite fields. In

particular, Reed-Muller code of order d is formed by all pn evaluations of

every polynomial over Fp with n variables of degree at most d. When d

becomes larger, the size of the code becomes larger while the minimum dis-

tance between codewords becomes smaller. Chapter 1 and Chapter 2 deal

with properties of degree-d polynomials, which in turn are easily interpreted

as properties of Reed-Muller codes of order d.

Small sample spaces cannot fool low degree polynomials

In Chapter 1 we study the minimal possible size of the support of a distri-

bution that ε-fools low degree polynomials over finite fields. A distribution

D ε-fools a set of functions F if for every f ∈ F , the distribution of f(x)

where x is taken according to D is ε-close to the distribution of f(x) where

x is taken according to the uniform distribution U . Constructions of such

distributions for various choices of F are widely used in theoretical computer

science.

We focus on the case where F is the set of degree-d polynomials over

Fnp . Construction for the case where d = 1 are well known as ε-biased sets,

and many works deal with such sets, including efficient constructions, lower

bounds and applications (see, for example, [6, 11, 12, 33, 95, 96, 103] and

their references). The existence of a small ε-biased set is equivalent to the

existence of a large linear code with large pairwise distances, and in fact

closing the gaps between the best lower and upper bounds on the best possible

size of an ε-bias set is essentially equivalent to the main open problem of the

theory of error correcting codes.

The case where d > 1 received a lot of attention in recent years. In

particular, construction of distributions that fool degree-d polynomials were

suggested in [39, 90, 109] and two applications of such sets appear in [92, 38].

We establish tight bounds on the minimum possible size of the support S of

Introduction 3

such a distribution, showing that any such S satisfies

|S| ≥ c1 ·(

( n2d

)d · log p

ε2 log (1ε)

+ p

).

This is nearly optimal as there is such an S of size at most

c2 ·(3nd

)d · log p+ p

ε2.

A key step in our proof is a new concentration bound on the bias of

degree-d multipartite polynomials. We use this concentration bound in order

to construct a large linear space of degree-d polynomials with small bias.

Observe that such a construction implies also the existence of a large subset of

the Reed-Muller code which forms a linear code with much higher minimum

distance. In Chapter 2 we improve this construction for polynomials over Fn2 .

References: The results of this chapter appear in:

• N. Alon, I. Ben-Eliezer and M. Krivelevich , Small sample spaces can-

not fool low degree polynomials,

The 12th International Workshop on Randomization and Com-

putation (RANDOM 2008), 266–275.

Random low degree polynomials are hard to approxi-

mate

In Chapter 2 we study the bias of a typical degree-d polynomial over Fn2 . The

bias of a polynomial p : Fn2 → F2 is defined as

bias (f) = Ex[(−1)f(x)

]= Pr [f (x) = 0]− Pr [f (x) = 1] .

That is, polynomials with small bias are more ’balanced’. We show that

there are some constants c1, c2 such that for a random degree-d polynomial

p we have

Pr[|bias (p)| > 2−c1n/d

]≤ 2−c2(

n≤d) ,

and we get the following consequences.

4 Introduction

• A new tail bound on the weight distribution of Reed-Muller codes.

• An improved construction of a linear space of degree-d polynomials

with small bias, which slightly improves the result of Chapter 1 for

polynomials over Fn2 .

• A typical degree-d polynomial cannot be approximated by any degree-

(d− 1) polynomial.

We note that an explicit construction of a degree-(d+ 1) polynomial that

cannot be approximated by degree d-polynomials will have a major impact in

theoretical computer science. Here we show that almost every degree-(d+ 1)

polynomial has this property.


• I. Ben-Eliezer, R. Hod and S. Lovett, Random low degree polynomials

are hard to approximate,

The 13th International Workshop on Randomization and Com-

putation (RANDOM 2009), 366–377.

Polynomial threshold functions: Structure and approx-

imation

While in Chapter 1 and Chapter 2 we study degree-d polynomials over finite

fields, in Chapter 3 we study polynomial threshold functions. These are sign

functions of real degree-d polynomials over the boolean cube. For example,

the Majority function can be expressed as a sign function of a linear real

polynomial. Recently, there has been a series of works studying properties

of polynomials threshold functions (see, e.g., [55, 56] and their references).

We prove that every low-degree polynomial threshold function admits a

relatively small partial assignment which results in a function with no very

influential variables. We then apply this result in order to prove that many

families of functions, including the family of low degree polynomials over

Introduction 5

finite fields, cannot be approximated by low-degree polynomial threshold

functions over the boolean cube.


• I. Ben-Eliezer, S. Lovett and A. Yadin, Polynomial Threshold Func-

tions: Structure, Approximation and Pseudorandomness.

Graphs

In Part II we present some extremal and probabilistic aspects of graphs and

digraphs. The first three chapters in this part focus on long paths and cycles

in directed graphs. In particular, we prove the existence of a long cycle

in every dense subgraph of a pseudorandom graph (Chapter 4), study the

minimal size of an oriented graph in which every edge-coloring contains a

long monochromatic path (Chapter 5) and study orientation games in which

one player tries to construct cycles and the second player tries to prevent it

(Chapter 6).

The next two chapters are devoted to the study of a Ramsey-tpe problem

that concerns rainbow colorings (Chapter 7) and to balanced partitions of

graphs (Chapter 8).

Long cycles in subgraphs of (pseudo)random directed

graphs

In Chapter 4 we study the resilience of pseudorandom directed graphs with

respect to the property of having a long directed cycle. In particular, our

results hold almost always for a graph distributed according to G(n, p) with

p = Ω( 1n).

We determine the function c(γ), such that for every pseudorandom graph

G the following holds. Every subgraphG′ ⊆ G that is obtained by a malicious

removal of at most (1/2− γ)-fraction of the edges, contains a directed cycle

6 Introduction

of length at least c(γ) · n. We also show that our result is tight up to low

order terms.

Our tools combine the Szemeredi’s regularity lemma for sparse directed

graphs (see [83]) with an elegant lemma that shows the existence of a long

directed path in expanding (bipartite) graphs using the well known DFS

algorithm. We use another variant of this lemma in Chapter 5.


• I. Ben-Eliezer, M. Krivelevich and B. Sudakov, Long cycles in sub-

graphs of (pseudo)random directed graphs,

Journal of Graph Theory, to appear.

The size Ramsey number of a directed path

In Chapter 5 we study the size Ramsey number of a directed path in oriented

graphs. The size Ramsey number re(H, q) is the minimal number m such that

there is a graph G with m edges with the property that every q-coloring of

E(G) contains a monochromatic copy of H.

An oriented graph is a directed graph with no parallel or anti parallel

edges. We provide nearly tight lower and upper bounds on re(Pn, q) for every

choice of q for oriented graphs. The size Ramsey number of Pn in general

(not oriented) directed graphs has been studied in [70, 100], and it was shown

that in this case re(Pn, 2) = Θ(n2). We prove that the size Ramsey number

in the oriented case is asymptotically larger.

The oriented case requires new techniques and ideas. In particular, we

prove the lower bound by showing that every oriented graph can be par-

titioned into not too many acyclic sets, where acyclic sets can be colored

without very long monochromatic paths. In the upper bound, we study fur-

ther properties of expansion in directed graphs, using a variant of the DFS

lemma as in Chapter 4.


Introduction 7

• I. Ben-Eliezer, M. Krivelevich and B. Sudakov, The size Ramsey num-

ber of a directed path.

Biased orientation games

In Chapter 6 we study biased orientation games. The board of the game is

the set of edges of the complete graph Kn. There are two players, Maker

and Breaker who take turns by orienting (or directing) previously undirected

edges. In the (p : q)-biased game, at each round, Maker directs at most

p edges and then Breaker directs at most q edges (usually, we consider the

case where p = 1 and q is large). By the end of the game, all the edges are

directed and we obtain a tournament T . Maker wins if T has some given

property P , and Breaker wins otherwise.

This game is a variant of the classical Maker-Breaker game, in which

Maker and Breaker take turns by claiming vertices of some hypergraph H,

and Maker wins if and only if he claims all the vertices of some hyperedge of

H.

The main goal is to determine for each property P the threshold bias

bP such that if b bP then Breaker wins the (1 : b)-biased game, and if

b bP then Maker wins the (1 : b)-biased game. We study the following

three properties.

• The cycle creation game. Maker wins if the obtained tournament

has a cycle while Breaker tries to prevent it (and construct an acyclic

tournament). We show if b ≤ (n/2 − 2) then Maker wins the game,

improving a bound of Bollobas and Szabo [44].

• The Hamiltonicity game. Maker wins if the obtained tournament

has a Hamilton cycle, and Breaker wins otherwise. We show that the

bias threshold for this property is (1+o(1))nlnn

.

• The H-creation game. Maker wins if the obtained tournament has a

copy of some fixed directed graph H, and Breaker wins otherwise. We

conjecture that the threshold bias for each H is closely related to the

8 Introduction

minimal feedback arc set of H, and provide some intermediate results

in this direction.


• I. Ben-Eliezer, M. Krivelevich and B. Sudakov, Biased orientation games.

Local rainbow colorings

Chapter 7 studies local rainbow colorings. Given a fixed graph H, we denote

by C(n,H) the minimum number k such that the following holds. There are

n colorings of E(Kn) with k-colors, each associated with one of the vertices

of Kn, such that for every copy T of H in Kn, at least one of the colorings

that are associated with V (T ) assigns distinct colors to all the edges of E(T ).

For example, it is rather easy to see that if H is a triangle then C(n,H) = 3.

We characterize the set of all graphs H for which C(n,H) is bounded by

some absolute constant c(H), and show that this set contains all graphs with

at most 3 edges, apart than P3, the path with 3 edges. We also provide a

general upper bound and some concrete lower bounds for graphs of special

interest.

As a special case, we prove that C(n, P3) = Ω(( lognlog logn

)1/4). This answers

a question raised by Wigderson [112] that is related to a linear-algebraic

computational model known as Span Programs.


• N. Alon and I. Ben-Eliezer, Local rainbow colorings.

Perfectly balanced partitions of smoothed graphs

Chapter 8 studies perfectly balanced partitions of graphs. For a graph G =

(V,E) of even order, a partition (V1, V2) of the vertices is said to be perfectly

balanced if |V1| = |V2| and the numbers of edges in the subgraphs induced

Introduction 9

by V1 and V2 are equal. Our goal is to prove that almost all graphs in some

sense admit a perfectly balanced partition.

For a base graph H define a random graph G(H, p) by turning every non-

edge of H into an edge and every edge of H into a non-edge independently

with probability p. We show that for any constant ε there is a constant α,

such that for any even n and a graph H on n vertices that satisfies ∆(H)−δ(H) ≤ αn, a graph G distributed according to G(H, p), with ε

n≤ p ≤ 1− ε

n,

admits a perfectly balanced partition with probability exponentially close to

1. As a direct consequence we get that for every p, a random graph from

G(n, p) admits a perfectly balanced partition with probability tending to 1.


• I. Ben-Eliezer and M. Krivelevich, Perfectly balanced partitions of

smoothed graphs.

Electronic Journal of Combinatorics, Vol. 16 (1), note N14, 2009.

In order to make every chapter self-contained, some basic definitions are

repeated in a few chapters.

10 Introduction

Part I

Codes and Polynomials

Chapter 1

Small sample spaces cannot

fool low degree polynomials

The results of this chapter appear in [8].

A distribution D on a set S ⊂ Fnp ε-fools polynomials of degree at most

d in n variables over Fp if for any such polynomial P , the distribution of

P (x) when x is chosen according to D differs from the distribution when x

is chosen uniformly by at most ε in the `1 norm. Distributions of this type

generalize the notion of ε-biased spaces and have been studied in several

recent papers. We establish tight bounds on the minimum possible size of

the support S of such a distribution, showing that any such S satisfies

|S| ≥ c1 ·(

( n2d

)d · log p

ε2 log (1ε)

+ p

).

This is nearly optimal as there is such an S of size at most

c2 ·(3nd

)d · log p+ p

ε2.

1.1 Introduction

Let P be a polynomial in n variables over Fp of degree at most d. Let D be

a distribution over a set S of vectors from Fnp , and denote by Un the uniform

14 Small sample spaces cannot fool low degree polynomials

distribution on Fnp . The distribution D is an ε-approximation of Un with

respect to P if∑a∈Fp

|Prx∼D [[P (x) = a]]− Prx∼Un [[P (x) = a]]| ≤ ε.

We say that S (with the distribution D) is an (ε, n, d)-biased space if it is

an ε-approximation with respect to any polynomial on n variables of degree

at most d. Note that D is not necessarily a uniform distribution over its

support S.

The case d = 1 is known as ε-biased spaces. Many works deal with such

spaces, including efficient constructions, lower bounds and applications (see,

for example, [6, 11, 12, 33, 95, 96, 103] and their references).

Luby et al. [92] gave an explicit construction for the general case, but

the size of their sample space S is 22O(√

log (n/ε))even for the case d = 2.

They used it to construct a deterministic approximation algorithm to the

probability that a given depth-2 circuit outputs a certain value on a random

input.

Bogdanov [37] gave better constructions that work for fields of size at

least poly(d, log n, 1ε). Bogdanov and Viola [39] suggested a construction for

general fields. The construction is the sum of d copies of ε′-biased spaces, and

the sample size is nd · f(ε, d, p) for some function f . However, the analysis of

their construction relies on the so called “Inverse Gowers Conjecture” which

was shown to be false [91]. Lovett [90] proved unconditionally for p = 2 that

the sum of 2d copies of ε′-biased spaces fools polynomials of degree d (where

ε′ is exponentially small in ε), thus giving an explicit construction of size

(nε)2O(d)

. Later, Viola [109] proved that the sum of d copies is sufficient. This

yields an explicit construction of size nd

εO(d·2d) using the best known construc-

tions of ε-biased spaces. Recently, Bogdanov et. al. [38] showed how to fool

width-2 branching programs using such distributions.

Here we study the minimum possible size of (ε, n, d)-biased spaces. Bog-

danov and Viola [39] observed that for p = 2 and ε < 2−d every such space

is of size at least(nd

). Their argument is very simple. The set of polynomials

of degree at most d forms a linear space of dimension∑d

i=0

(ni

)>(nd

). If

S is of size less than(nd

)then there is a non-zero polynomial P such that

1.2 Proofs 15

P (x) = 0 for every x ∈ S, and since every non zero polynomial is not zero

with probability at least 2−d (as follows, for example, by considering the min-

imal distance of the Reed-Muller code of order d) we get the desired bound.

However, their bound doesn’t depend on ε and, for small values of ε, is far

from optimal and also from the known bound for ε-biased space, which is

nearly optimal for d = 1. Our main contribution is a nearly tight lower

bound on the size of such spaces as a function of all four parameters ε, n,d

and p. Note that as spaces of this type can be useful in derandomization,

where the running time of the resulting algorithms is proportional to the size

of the space, it is interesting to get a tight bound for their smallest possible

size.

Theorem 1.1. There exists an absolute constant c1 > 0 such that for every

d ≤ n10

and ε ≥ d ·p− n2d , every (ε, n, d)-biased space over Fp has size at least

max

c1 ·

( n2d

)d log p

ε2 log (1ε), p(1− ε)

.

We also observe that this bound is nearly tight by proving the following

simple statement.

Proposition 1.2. There is an absolute constant c2 > 0 so that for every

d ≤ n10

there is an (ε, n, d)-biased space over Fp of size at most c2 ·( 3nd

)d log p+p

ε2.

The proofs are described in the next section, and they use a slight variant

of a bound on the rank of perturbed identity matrices that was presented

in [5] . For completeness, we prove this variant in Section 1.3. The final

section contains some concluding remarks. Throughout the proofs we omit

all floor and ceiling signs whenever these are not crucial.

1.2 Proofs

In this section we present the proofs of our results. The proof of our main

result, Theorem 1.1, lower bounding the size of an (ε, n, d)-biased set, is given

in Section 1.2.1. The proof of the upper bound (Proposition 1.2) is in Section

1.2.2.


1.2.1 Lower bound

First we observe that a bound of p(1 − ε) follows easily as otherwise the

distribution doesn’t fool every polynomial P for which P (x) is the uniform

distribution (for example, all the linear polynomials). Let n be the number

of variables and let d be the degree of the polynomial. Assume for simplicity

that n = Nd, where N is an integer. For every i ≥ 1 define the set of

variables Si = xi,1, ..., xi,N. A monomial over Fp is called d-partite if it has

the form∏

1≤i≤d xi,ji , and a polynomial over Fp is called d-partite if it is a sum

of d-partite monomials. Note that d-partite polynomials are homogeneous

polynomials of degree d.

Let PN,d be the uniform distribution on the set of d-partite polynomials.

A random element in PN,d is a sum of d-partite monomials, where every one

of the possible Nd monomials has a random coefficient selected uniformly

and independently from Fp.An assignment to the variables xi is non-trivial if there is an i such

that xi 6= 0. Similarly, if v1, v2, ..., vn ∈ V for some vector space V , a linear

combination∑

i αivi is non-trivial if there is i such that αi 6= 0. For a prime

p, a polynomial P over Fp is δ-balanced if∑a∈Fp

∣∣∣∣ |x : P (x) = a|pn

− 1

p

∣∣∣∣ ≤ δ.

A polynomial is balanced if it is 0-balanced.

Note that for p = 2, we define the bias of a polynomial as Ex[(−1)f(x)

].

Therefore, a polynomial P over F2 is δ-balanced if and only if it is 2δ-biased.

However, for the sake of simplicity in this chapter we will use the notion of

balanced polynomials. In Chapter 2, which focuses on polynomials over F2,

we defer the use of the more standard bias notion.

We have the following key lemma.

Lemma 1.3. The probability φ(N, d) that a random element from PN,d is

d · p−N2 -balanced is at least

1− p−(N2

)d+2(N2

)d−1+∑d−3i=0 (N

2)i(N

2

4+N) ≥ 1− p−(N

2)d+4(N

2)d−1

.

1.2 Proofs 17

Proof. We apply induction on d. For d = 1, as every non-trivial linear

polynomial is balanced, we have φ(N, 1) = 1−p−N , and the statement holds.

Assuming that the statement is valid for d, we prove it for d+ 1. A random

(d + 1)-partite polynomial P can be represented as∑N

i=1 x1,iPi, where for

every i, Pi is a random polynomial (distributed uniformly and independently

over PN,d) over the sets of variables S2, S3, ..., Sd+1. Denote the set Pi of

polynomials by P . We use the following claim.

Claim 1.4. With probability at least 1− p−(N2

)d+1+2(N2

)d+∑d−1i=0 (N

2)i(N

2

4+N) over

the choice of polynomials in P, there is a subset B ⊆ P of size at least N2

such that for any non-trivial choice of αi, the polynomial∑

Pi∈B αiPi is

d · p−N2 -balanced.

Proof. Let B0 := ∅. In the i’th step, we consider the polynomial Pi. If Pias well as all its combinations with elements from Bi−1 are d · p−N2 -balanced,

we set Bi := Bi−1

⋃Pi, otherwise we call the step bad and let Bi := Bi−1.

After the last polynomial, set B := BN . We want to bound the probability

that there are more than N2

bad steps. Consider a certain step i and assume

that |Bi−1| < N2

. Since Pi is a random polynomial, the sum of Pi with

every fixed polynomial is uniformly distributed over the set PN,d. By the

induction hypothesis, it is d · p−N2 -balanced with probability at least 1 −p−(N

2)d+2(N

2)d−1+

∑d−2i=0 (N

2)i(N

2

4+N). By the union bound, the probability that

the step is bad is at most

pN/2 · p−(N2

)d+2(N2

)d−1+∑d−2i=0 (N

2)i(N

2

4+N).

We bound the probability that there are more than n2

bad steps. For d = 2

the probability is at most(NN2

)(pN/2 · p−N

)N/2 ≤ p−(N2

)2+N .

For d ≥ 3, we have(NN2

)(pN/2 · p−(N

2)d+2((N

2)d−1)+

∑d−2i=0 (N

2)i(N

2

4+N))N/2

≤ p−(N2

)d+1+2((N2

)d)+∑d−1i=0 (N

2)i(N

2

4+N).

The claim follows.


Assume that the condition of the claim holds, and without loss of general-

ity assume that P1, P2, ..., PN/2 ⊆ B. Let P ′ =∑N/2

i=1 x1,iPi. By Claim 1.4,

for every non-trivial assignment of the variables x1,i, the obtained polyno-

mial is d ·p−N2 -balanced. The probability that the assignment of the variables

x1,i is trivial is p−N2 . Therefore, P ′ is δ-balanced, where

δ ≤ p−N2 + d · p−

N2 = (d+ 1) · p−

N2 . (1.1)

We use this fact to prove that the polynomial P is (d+1) ·p−N2 -balanced.

For every assignment of the variables from⋃

2≤i≤d+1 Si, P reduces to a linear

polynomial, which depends only on the variables from S1. Denote by µ(P )

(respectively, µ(P ′)) the probability over the assignments of⋃

2≤i≤d+1 Si that

P (respectively, P ′) reduces to a trivial linear polynomial. Clearly µ ≤ µ′

and µ is an upper bound on the imbalance of P . Therefore, it is sufficient to

prove that µ′ is bounded by (d + 1) · p−N2 . To this end, note that whenever

P ′ is reduced to a constant polynomial it is actually reduced to the zero

polynomial. Therefore, as the bias of P ′ is bounded by (d + 1) · p−N2 , the

lemma follows.

We construct a set of polynomials Q as follows. Let

r = logp (1

1− φ(N, d))− 1 ≥ (

n

2d)d − 4(

n

2d)d−1 − 1.

For every 1 ≤ i ≤ r let qi be a polynomial distributed uniformly and inde-

pendently over PN,d. Denote by Q the set of all non-trivial combinations of

q1, ..., qr.By the union bound and by Lemma 1.3, with positive probability all

the elements of Q are d · p−N2 -balanced. Fix Q to be such a set. It fol-

lows also that the vectors q1, q2, ..., qr are linearly independent (otherwise

Q contains the zero vector, which is not d · p−N2 -balanced). Therefore,

|Q| ≥ p( n2d

)d−4( n2d

)d−1−1 − 1.

The following lemma is due to Alon [5].

Lemma 1.5 ([5]). There exists an absolute positive constant c so that the

following holds. Let B be an n by n real matrix with bi,i ≥ 12

for all i and

1.2 Proofs 19

|bi,j| ≤ ε for all i 6= j where 12√n≤ ε ≤ 1

4. Then the rank of B satisfies

rank(B) ≥ c log n

ε2 log (1ε).

Here we need the following complex variant of the lemma.

Lemma 1.6. There exists an absolute positive constant c so that the following

holds. Let C be an n by n complex matrix with |ci,i| ≥ 12

for all i and |ci,j| ≤ ε

for all i 6= j where 12√n≤ ε ≤ 1

4. Then the rank of C satisfies

rank(C) ≥ c log n

ε2 log (1ε).

We give the proof of this variant in Section 1.3. For completeness we also

reproduce there the proof of Lemma 1.5.

We are now ready to prove Theorem 1.1.

Proof. Theorem 1.1 Suppose that W is an (ε, n, d)-biased space, and that

W = w1, w2, ..., wm, Pr [[wi]] = ti. Define a |Q|-by-m complex matrix U

whose rows are indexed by the elements of Q and whose columns are indexed

by the elements of W . Set Uq,wi = (ξp)q(wi)√ti, where ξp is a primitive root

of unity of order p and the value of q(wi) is computed over Fp. Note that

by our choice of Q and the definition of an (ε, n, d)-biased space, for every

q ∈ Q we have

|m∑i=1

(ξp)q(wi) · ti| ≤ ε+ d · p−

N2 ≤ 2ε.

Also, obviouslym∑i=1

ti = 1.

For every two distinct polynomials q1, q2 ∈ Q, the polynomial q1 − q2 is

also in Q, and for every wi we have

(ξp)(q1−q2)(wi) = (ξp)

q1(wi) · (ξp)−q2(wi).


Set A = UU∗. For every distinct q1, q2 ∈ Q we have

|Aq1,q2| = |m∑i=1

(ξp)(q1−q2)(wi) · ti| ≤ 2ε.

All the diagonal entries in A are 1. Since the rank of U is at most m the

rank of A is also at most m. By Lemma 1.6 we get that

m ≥ rank(A) ≥ c′ · log |Q|ε2 log (1

ε)≥ c1 ·

( n2d

)d · log p

ε2 log (1ε).

The desired result follows.

1.2.2 Upper bound

Here we prove the simple upper bound.

Proof. Proposition 1.2 LetR ⊆ Fnp be a random set of sizem = 2 · ( 3nd

)d log (p)+p

ε2.

We bound the probability that for a given polynomial P , the uniform distri-

bution on R is not an ε-approximation with respect to P .

Let L ⊂ Fp, and let µL = m∑

a∈L Prx∈Un [[p(x) = a]] be the expected

number of vectors from R such that P evaluates to elements from L. By the

Chernoff bounds (see, e.g., [13], Appendix A), we have

PrR [[Prx∈Un [[P (x) ∈ L]]− Prx∈R [[P (x) ∈ L]] > ε]] ≤ e−µL·( εmµL )2/2 ≤ e−mε

2/2.

By the union bound over all 2p possible sets L, the probability that the

uniform distribution on R is not an ε-approximation is at most e−mε2/2+p.

The number of normalized monomials of degree at most d is exactly the

number of ways to put d identical balls in n+1 distinct bins, and is bounded

by (d+ n

d

)≤(e(n+ d)

d

)d≤(

3n

d

)d.

Therefore the total number of polynomials of degree at most d is at most

p( 3nd

)d = 2( 3nd

)d·log p.

By applying the union bound, with high probability the uniform distri-

bution on R is an ε-approximation with respect to any polynomial on n

variables with degree at most d, and the theorem follows.

1.3 A complex variant of Lemma 1.5 21

1.3 A complex variant of Lemma 1.5

In this section we reproduce the proof of Lemma 1.5 (omitting the final

detailed computation) as given in [5], and also prove Lemma 1.6.

We start with the following lemma from which Lemma 1.5 will follow.

Lemma 1.7. There exists an absolute positive constant c so that the following

holds. Let B be an n by n real matrix with bi,i = 1 for all i and |bi,j| ≤ ε for

all i 6= j. If 1√n≤ ε < 1/2, then

rank(B) ≥ c

ε2 log(1/ε)log n.

We need the following well known lemma proved, among other places, in

[52], [4].

Lemma 1.8. Let A = (ai,j) be an n by n real, symmetric matrix with ai,i = 1

for all i and |ai,j| ≤ ε for all i 6= j. If the rank of A is d, then

d ≥ n

1 + (n− 1)ε2.

In particular, if ε ≤ 1√n

then d > n/2.

Proof. Let λ1, . . . , λn denote the eigenvalues of A, then their sum is the trace

of A, which is n, and at most d of them are nonzero. Thus, by Cauchy-

Schwartz,∑n

i=1 λ2i ≥ d(n/d)2 = n2/d. On the other hand, this sum is the

trace of AtA, which is precisely∑

i,j a2i,j ≤ n+ n(n− 1)ε2. Hence n+ n(n−

1)ε2 ≥ n2/d, implying the desired result.

Lemma 1.9. Let B = (bi,j) be an n by n matrix of rank d, and let P (x)

be an arbitrary polynomial of degree k. Then the rank of the n by n matrix

(P (bi,j)) is at most(k+dk

). Moreover, if P (x) = xk then the rank of (P (bi,j))

is at most(k+d−1k

).

Proof. Let v1 = (v1,j)nj=1,v2 = (v2,j)

nj=1, . . . ,vd = (vd,j)

nj=1 be a basis of the

row-space of B. Then the vectors (vk11,j · vk22,j · · · v

kdd,j)

nj=1, where k1, k2, . . . , kd

range over all non-negative integers whose sum is at most k, span the rows

of the matrix (P (bi,j)). In case P (x) = xk it suffices to take all these vectors

corresponding to k1, k2, . . . , kd whose sum is precisely k.


Proof. Lemma 1.7 We may and will assume that B is symmetric, since other-

wise we simply apply the result to (B+Bt)/2 whose rank is at most twice the

rank of B. Put d = rank(B). If ε ≤ 1/nδ for some fixed δ > 0, the result fol-

lows by applying Lemma 1.8 to a b 1ε2c by b 1

ε2c principal submatrix of B. Thus

we may assume that ε ≥ 1/nδ for some fixed, small δ > 0. Put k = b logn2 log(1/ε)

c,n′ = b 1

ε2kc and note that n′ ≤ n and that εk ≤ 1√

n′. By Lemma 1.9 the rank

of the n′ by n′ matrix (bki,j)i,j≤n′ is at most(d+kk

)≤ ( e(k+d)

k)k. On the other

hand, by Lemma 1.8, the rank of this matrix is at least n′/2. Therefore(e(k + d)

k

)k≥ n′

2=

1

2b 1

ε2kc,

and the desired result follows by some simple (though somewhat tedious)

manipulation, which we omit.

Proof. Lemma 1.5 Let C = (ci,j) be the n by n diagonal matrix defined by

ci,i = 1/bi,i for all i. Then every diagonal entry of CB is 1 and every off-

diagonal entry is of absolute value at most 2ε. The result thus follows from

Lemma 1.7.

Proof. Lemma 1.6 Let P be an n by n diagonal matrix defined by pi,i = 1/ci,iand set D = CP . Then every diagonal entry of D is 1 and every off-diagonal

entry is of absolute value at most 2ε. Set D′ = (D+D∗)/2. Then D′ is a real

matrix and rank(D′) ≤ 2 · rank(D). The desired result follows by applying

Lemma 1.7 to D′.

1.4 Concluding Remarks

For p(nd

), the ratio between the lower and upper bounds is c · (2e)d log (1

ε)

for some constant c. In particular, for fixed d the ratio is Θ(log (1ε)). This

matches the ratio between the best known upper and lower bounds in the

case d = 1 that corresponds to ε-biased spaces. In Chapter 2 we present a

subsequent work that studies the bias of a random degree-d polynomial over

Fn2 . The results there improve the lower bound for the case that p = 2 by a

factor of (2e)d.

1.4 Concluding Remarks 23

Our bound is valid only for ε ≥ d · p− n2d . As noted in [5], for ε ≤ p−

n2

every ε-biased space must be essentially the whole space (even for d = 1). It

may be interesting to close the gap between p−n2 and d · p− n

2d .

Schechtman and Shraibman [102] proved a strengthening of Lemma 1.5.

They showed that under the conditions of Lemma 1.5, if A is also positive-

semidefinite then we need only an upper bound on the values of non-diagonal

entries, instead of an upper bound on their absolute values. In our case, for

p = 2 the matrix A is positive semidefinite, and we can thus relax the condi-

tions and establish a similar lower bound for the size of the support of any

distribution in which no polynomial attains the value zero with probability

bigger by ε/2 than the probability it attains it in the uniform distribution.

That is, for p = 2 the lower bound for the size of the distribution holds, even

if there is no lower bound on the probability that each polynomial attains

the value zero.

Lemma 1.3 can also be formulated in the language of error correcting

codes. For given n and d, it states that every Reed-Muller code with pa-

rameters n and d contains a dense linear subcode in which every nontrivial

codeword is balanced.

Dvir and Shpilka [58] gave an efficient encoding and decoding procedures

for the construction of sum of d copies of ε-biased spaces.


Chapter 2

Random low degree

polynomials are hard to

approximate


We study the problem of how well a typical multivariate polynomial can

be approximated by lower degree polynomials over F2. We prove that almost

all degree d polynomials have only an exponentially small correlation with all

polynomials of degree at most d− 1, for all degrees d up to Θ (n). That is, a

random degree d polynomial does not admit a good approximation of lower

degree. In order to prove this, we prove far tail estimates on the distribution

of the bias of a random low degree polynomial. Recently, several results

regarding the weight distribution of Reed–Muller codes were obtained. Our

results can be interpreted as a new large deviation bound on the weight

distribution of Reed–Muller codes.

2.1 Introduction

Two functions f, g : Fn2 → F2 are said to be ε-correlated if

Pr [f(x) = g(x)] ≥ 1 + ε

2.

26 Random low degree polynomials are hard to approximate

A function f : Fn2 → F2 is said to be ε-correlated with a set of functions

F ⊆ Fn2 → F2 if it is ε-correlated with at least one function g ∈ F .

We are interested in functions that have a low correlation with the set

of degree d− 1 polynomials; namely, functions that cannot be approximated

by any polynomial of total degree at most d − 1. How complex must such

a function be? We use the most natural measure for complexity in these

settings, which is the degree of the function when considered as a polynomial.

A simple probabilistic argument shows that for any constant δ < 1 and

for d < δn, a random function has an exponentially small correlation with

degree d−1 polynomials. However, a random function is complex since, with

high probability, its degree is at least n − 2. In this chapter, we study how

well a random degree d polynomial can be approximated by any lower degree

polynomial, and show that with very high probability a random polynomial

of degree d cannot be approximated by polynomials of lower degree in a

strong sense. Thus, if we want to find functions that are uncorrelated with

degree d− 1 polynomials, considering degree d polynomials is enough.

2.1.1 Motivation

The correlation of a typical degree d polynomial with the set of lower degree

polynomials is a natural question in arithmetic complexity. More generally,

the study of the correlation of functions with the set of low degree polynomials

is interesting from both coding theory and complexity theory points of view.

Complexity Theory. Approximation of functions by low degree polyno-

mials is one of the major tools used in proving lower bounds for constant

depth circuits. For example, Razborov and Smolensky [99, 104] provided an

explicit function Mod3 that cannot be computed by a constant depth cir-

cuit with a subexponential number of And, Or and Xor gates. The proof

combines the following two arguments.

1. Any constant depth circuit of subexponential size has a very high cor-

relation (that is, 1− o (1)) with some polynomial of degree poly log n;

2.1 Introduction 27

2. Such a low degree polynomial has a correlation of at most 2/3 with

Mod3. (In fact, this is true for any polynomial of degree at most ε√n

for some constant ε.)

The best known constructions of explicit functions that cannot be approxi-

mated by low degree polynomials (see, e. g., [32, 16, 99, 104, 111]) fall into

two categories.

• For large degree bounds (d < nΩ(1)), there exists a symmetric function

with a correlation of at most O (1/√n) with degree O (

√n) polynomi-

als;

• For small degree bounds (d < log n) there are explicit functions having

a correlation of at most exp(−n/cd) with degree d polynomials for some

constants c (best known is c = 2.)

Certain applications, e. g., pseudorandom generator constructions via the

Nisan–Wigderson construction [97], require a function having an exponen-

tially small correlation with low degree polynomials. This is only known for

degrees up to log n, while for larger degrees the best known bound is polyno-

mial in n. Finding explicit functions with a better correlation is an ongoing

quest with limited success. For more details, see a survey by Viola [110].

Coding Theory. The Reed–Muller code RM (n, d) is a linear code in

which codewords correspond to polynomials (over F2) in n variables of total

degree at most d. This family of codes is one of the most studied objects in

coding theory (see, e.g., [93]). Nevertheless, determining the weight distribu-

tion of these codes (for d ≥ 3) is a long standing open problem. Interpreted

in this language, our main lemma gives a new tail estimate on the weight

distribution of Reed–Muller codes.

2.1.2 Our results

We show that, with very high probability, a random degree d polynomial has

an exponentially small correlation with polynomials of lower degree, i.e. of

degree at most d− 1. We prove this for degrees ranging from a constant up


to δmaxn, where 0 < δmax < 1 is an absolute constant. All results hold for

large enough n.

Theorem 2.1 (Main Theorem). There exist constants 0 < δmax < 1 and

c, c′ > 0 such that the following holds. for every d ≤ δmaxn let f be a

random n-variate polynomial of degree d. Then the probability that f has a

correlation 2−cn/d with polynomials of degree at most d−1 is at most 2−c′( n≤d),

where(n≤d

)=∑d

i=0

(ni

).

The main theorem is an easy corollary of the following lemma, which

is the main technical contribution of this chapter. We define the bias of a

function f : Fn2 → F2 to be

bias (f) = Ex[(−1)f(x)

]= Pr [f (x) = 0]− Pr [f (x) = 1] .

In Chapter 1 we defined the notion of balanced polynomials, which is es-

sentially the same as the bias of polynomials for the case of F2. We proved

there that a random d-partite polynomials has small bias with very high

probability. Our main lemma extends this result for random degree-d poly-

nomials over F2.

Lemma 2.2 (Main Lemma). Fix ε > 0 and let f be a random degree d

polynomial for d ≤ (1− ε)n. Then,

Pr[|bias (f)| > 2−c1n/d

]≤ 2−c2(

n≤d) ,

where 0 < c1, c2 < 1 are constants depending only on ε.

Note that Lemma 2.2 holds for degrees up to (1− ε)n, while we were only

able to prove Theorem 2.1 for degrees up to δmaxn. The proof of Lemma 2.2

appears in Section 2.2.1.

The following proposition (proved in Section 2.3) shows that the estimate

in Lemma 2.2 is somewhat tight for degrees up to n/2.

Proposition 2.3. Fix ε > 0 and let f be a random degree d polynomial

for d ≤ (1/2− ε)n. Then,

Pr[|bias (f)| > 2−c

′1n/d]≥ 2−c

′2(

n≤d) ,

where 0 < c′1, c′2 < 1 are constants depending only on ε.

2.1 Introduction 29

Our proof of Lemma 2.2 uses the following tight lower bound on the

dimension of truncated Reed–Muller codes, which has appeared in [82, The-

orem 1.5]. For the sake of self containment, we present an alternative proof

of Lemma 2.4. Our proof, unlike the original, has an algorithmic flavor.

Lemma 2.4. Let x1, . . . , xR be R = 2r distinct points in Fn2 . Consider the

linear space of degree d polynomials restricted to these points; that is, the

space

(p (x1) , . . . , p (xR)) : p ∈ RM (n, d) .

The linear dimension of this space is at least(r≤d

).

Large spaces of polynomials with small bias and (ε, n, d)-biased

spaces. Recall that in Chapter 1 we proved a tail estimate similar to

Lemma 2.2 for d-partite polynomials over Fp. We used it in order to con-

struct a large linear space of degree-d polynomials with small bias. Combin-

ing Lemma 2.2 (instead of Lemma 1.3 from Chapter 1) and a simple union

bound gives the following. There is a linear space with dimension Θ((n≤d

)) of

degree-d polynomials over Fn2 , each with bias at most 2−c1n/d. Following the

proof of the lower bound in Chapter 1 yields that if S is an (ε, n, d)-biased

space then

|S| ≥ Ω

( (nd

)ε2 log (1/ε)

),

This improves the lower bound on the size of (ε, n, d)-biased spaces over F2

by a factor of roughly (2e)d.

2.1.3 Related Work

The weight distribution of Reed–Muller codes is completely known for d = 2

(see, for example, [53]) and some partial results are known also for d = 3.

In the general case, there are estimates (see, e.g., [78, 79]) on the number

of codewords with weight between w and 2.5w, where w = 2−d is the mini-

mal weight of the code. Kaufman and Lovett [81] proved bounds for larger

weights, and following Gopalan et al. [67], they used it to prove new bounds

for the list-decoding of Reed–Muller codes.


The Gowers Norm is a measure related to the approximability of functions

by low degree polynomials. It was introduced by Gowers [68] in his seminal

work providing a new proof for Szemerdi’s Theorem. Using the Gowers

Norm machinery, it is easy to prove that a random polynomial of degree

d < log n has a small correlation with lower degree polynomials. However,

this approach fails for degrees exceeding log n. In contrast, note that our

result holds for degrees up to δmaxn.

Green and Tao [69] study the structure of biased multivariate polynomi-

als. They prove that if their degree is at most the size of the field (which

in our case is 2), then they must have structure — they can be expressed

as a function of a constant number of lower degree polynomials. Kaufman

and Lovett [80] strengthen this structure theorem for polynomials of every

constant degree, removing the field size restriction.

2.2 Proof of the Main Theorem

First we show that Theorem 2.1 follows directly from Lemma 2.2 by a simple

counting argument.

Let f be a random degree d polynomial for d ≤ δmaxn, where δmax will

be determined later. For every polynomial g of degree at most d− 1, f − gis also a random degree d polynomial. By the union bound for all possible

choices of g,

Prf[∃g ∈ RM(n, d− 1) : |bias (f − g)| ≥ 2−c1n/d

]≤ 2( n

≤d−1)−c2(n≤d)

Choosing δmax to be a small enough constant, we get that there is a con-

stant c′ > 0 such that c2

(n≤d

)−(

n≤d−1

)≥ c′

(n≤d

)for all d ≤ δmaxn (see, for

example, [75, Exercise 1.14]).

We now move on to prove Lemma 2.2. The rest of this section is or-

ganized as follows. Lemma 2.2 is proved in Subsection 2.2.1, where the

technical claims are postponed to Subsection 2.2.2. Lemma 2.4 is proved in

Subsection 2.2.3.

2.2 Proof of the Main Theorem 31

2.2.1 Proof of Lemma 2.2

We need to prove that a random degree d polynomial has a very small bias

with very high probability. Denote byRM (n, d)⊥ the dual code ofRM (n, d).

We start by correlating the moments of the bias of a random degree d poly-

nomial to short words in RM (n, d)⊥.

Claim 2.5. Fix t ∈ N and let p ∈ RM (n, d) and x1, . . . , xt ∈ Fn2 be chosen

independently and equiprobably. Then,

E[bias(p)t

]= Pr

[ex1 + · · ·+ ext ∈ RM(n, d)⊥

],

where ex for x ∈ Fn2 is the unit vector in F2n

2 , having 1 in position x and 0

elsewhere.

In favor of not interrupting the proof, we postpone the proof of Claim 2.5

and other technical claims to Subsection 2.2.2.

We proceed by introducing the following definitions. Fix d. For x ∈ Fn2let evald(x) denote its d-evaluation; that is, a (row) vector in F( n

≤d)2 whose

coordinates are the evaluation of all monomials of degree up to d at the

point x. Formally,

evald(x) =

(∏i∈I

x(i)

)I⊂[n],|I|≤d

.

For points x1, . . . , xt ∈ Fn2 let Md(x1, . . . , xt) denote their d-evaluation ma-

trix ; this is a t ×(n≤d

)matrix whose i’th row is the d-evaluation of xi. We

denote the rank of Md(x1, . . . , xt) by rankd(x1, . . . , xt). As this value is in-

dependent of the order of x1, . . . , xt, we may refer without ambiguity to the

d-rank of a set S ⊆ Fn2 by rankd(S).

According to Claim 2.5, in order to bound the moments of the bias of a

random polynomial we need to study the probability that a random word of

length about1 t is in RM (n, d)⊥.

Let A = Md(x1, . . . , xt). Note that ex1 + · · · + ext ∈ RM(n, d)⊥ if and

only if ex1 + · · ·+ ext is orthogonal to all degree d polynomials, namely, if

p (x1) + · · ·+ p (xt) = 0 (2.1)

1We say “about t” as x1, . . . , xt might not be distinct.


for any degree d polynomial p. It is sufficient to satisfy (2.1) only on the

monomial basis of the degree d polynomials; that is, verify that each column

in A sums to zero. Therefore, ex1 + · · ·+ ext ∈ RM(n, d)⊥ if and only if the

sum of the rows of A is zero.

We turn to bound the probability that the rows of A sum to the zero

vector for random x1, . . . , xt ∈ Fn2 . For this we divide the n variables into

two sets: V ′ of size n′ = dn(1− 1/d)e and V ′′ of size n′′ = n − n′. Let

α = n′′/n ≈ 1/d. Instead of requiring that every column of A sums to zero,

we require this only for columns corresponding to monomials that contain

exactly one variable from V ′′ (and thus up to d− 1 variables from V ′).

For i = 1, . . . , t denote by x′i (∈ Fn′2 ) the restriction of xi ∈ Fn2 to the

variables in V ′. The following claim bounds the probability that sum of A’s

rows is zero in terms of the (d− 1)-rank of x′1, . . . , x′t.

Claim 2.6.

Prxi[ex1 + · · ·+ ext ∈ RM(n, d)⊥

]≤ Ex′i

[2−rankd−1(x′1,...,x

′t)αn].

To finish the proof, we provide a (general) lower bound on d-ranks of

random vectors.

Claim 2.7. For all fixed β < 1 and δ < 1, there exist constants c > 0 and

η > 1 such that if x1, . . . , xt ∈ Fn2 are chosen uniformly and independently,

where t ≥ η(n≤d

)and d ≤ δn, then

Pr

[rankd(x1, . . . , xt) < β

(n

≤ d

)]≤ 2−c(

n≤d+1) .

We now put it all together, in order to complete the proof of Lemma 2.2.

According to Claim 2.6, we have

Prxi[ex1 + · · ·+ ext ∈ RM(n, d)⊥

]≤ Ex′i

[2−rankd−1(x′1,...,x

′t)αn].

Applying Claim 2.7 for d − 1 and n′ (instead of d and n in the claim

statement), and assuming t ≥ η(

n′

≤d−1

), we get that

Pr

[rankd−1(x′1, . . . , x

′t) < β

(n′

≤ d− 1

)]< 2−c(

n′≤d) .


Therefore,

Prxi[ex1 + · · ·+ ext ∈ RM(n, d)⊥

]≤ 2−β(

n′≤d−1)αn + 2−c(

n′≤d) .

Recalling that n′ = dn(1− 1/d)e and α = 1 − n′/n = 1/d + O(1/n), we

get that for any constant β (and c = c(β)) there is a constant c′ such that

Prxi[ex1 + · · ·+ ext ∈ RM(n, d)⊥

]≤ 2−c

′( n≤d) .

This is because(

n′

≤d−1

)= Θ

((n≤d

)d/n)

and(n′

≤d

)= Θ

((n≤d

)).

We thus proved that there is a constant c′ such that

Ef∈RM(n,d)

[bias(f)t

]≤ 2−c

′( n≤d) ,

for t = η(

n′

≤d−1

)= Θ

((n≤d−1

)). Hence, tn/d ≤ c′′

(n≤d

)for some constant c′′.

For small enough c1 > 0 such that c2 = c′−c′′c1 > 0, by Markov inequality,

Pr[|bias(f)| ≥ 2−c1n/d

]≤ 2tc1n/d−c

′( n≤d) ≤ 2(c′′c1−c′)( n

≤d) ≤ 2−c2(n≤d) .

2.2.2 Proofs of technical claims

Proof of Claim 2.5. Write p as

p(x) =∑

I⊂[n],|I|≤d

αI∏i∈I

x(i) ,


where x(i) denotes the ith coordinate of x ∈ Fn2 . As p was chosen uniformly,

all αI are uniform and independent over F2. Therefore,

Ep[(bias(p))t

]= Ep

[t∏

j=1

bias(p)

]

= EαI

[t∏

j=1

Exj[(−1)

∑I αI

∏i∈I xj(i)

]]

= Exj

[∏I

EαI[(−1)αI(

∑tj=1

∏i∈I xj(i))

]]

= Exj

[∏I

1∑tj=1

∏i∈I xj(i)=0

]

= Prxj

[∀I

t∑j=1

∏i∈I

xj(i) = 0

]= Prxj

[ex1 + · · ·+ ext ∈ RM(n, d)⊥

].

Proof of Claim 2.6. Let A′ =Md−1(x′1, . . . , x′t) be the t×

(n′

≤d−1

)sub-matrix

of A corresponding to monomials of degree at most d−1 in variables from V ′.

Let E be the event in which every column of A corresponding to a monomial

that contains exactly one variable from V ′′ sums to zero.

We observe that this event is equivalent to the event that every column

of A′ is orthogonal to the set of vectors (x1(i), . . . , xt(i)) : i ∈ V ′′, as the

inner product of every column from A′ with vectors from this set corresponds

to a monomial of degree at most d.

Fix the variables in V ′; this determines A′. As the variables in V ′′ are

independent of those in V ′, the probability of E (given A′) is(2−rank(A′)

)|V ′′| = 2−rank(A′)αn = 2−rankd−1(x′1,...,x

′t)αn .

This holds for every assignment for variables of V ′, hence the result follows.


Proof of Claim 2.7. Let B = Md(x1, . . . , xt) be the t ×(n≤d

)d-evaluation

matrix of the random x1, . . . , xt ∈ Fn2 . We need to bound the probability

that rank(B) < β(n≤d

).

Fix some b ≤ β(n≤d

), and let us consider the event that the first b rows

of B span the entire row span of B. Denote by V the linear space spanned

by the first b rows of B. Since all rows of B are d-evaluations of some points

in Fn2 , we need to study the maximum number of d-evaluations contained in

a linear subspace of dimension b.

Assume there are at least 2r distinct d-evaluations in V . By Lemma 2.4,

dim(V ) ≥(r≤d

). Assume further that rank(B) < β

(n≤d

); we get that

β

(n

≤ d

)> rank(B) ≥ dim(V ) ≥

(r

≤ d

).

By Claim 2.8, r ≤ n(1−γ/d), where γ is a constant depending only on β. In

other words, out of the 2n d-evaluations of all points in Fn2 , at most 2n(1−γ/d)

fall in V and hence the probability that a random d-evaluation is in V is at

most 2−γn/d.

Assume without loss of generality that the number of rows t is exactly

η(n≤d

)for some η > 1. The probability that all the remaining rows of B are

in V is at most(2−γn/d

)t−b ≤ 2−(η−β)( n≤d)γn/d ≤ 2−γρ(η−β)( n

≤d+1) ,

where the last inequality follows from the fact that there exists a constant

ρ > 0 such that (n/d)(n≤d

)≥ ρ(

n≤d+1

)for all n and d.

Choosing η > β (and large enough n), we get that when we union bound

over all possible ways to choose at most β(n≤d

)rows out of t = η

(n≤d

), the

probability that any of them spans the rows of B is at most 2−c(n≤d+1), where c

depends only on β.

Claim 2.8. For any β, δ < 1, there is a constant γ = γ(β, δ) such that if

1 ≤ d ≤ δn and r ≥ d satisfy β(n≤d

)≥(r≤d

)then r ≤ n(1− γ/d).

Proof. We bound

1

β≤(n≤d

)(r≤d

) ≤ max0≤i≤d

(ni

)(ri

) =

(nd

)(rd

) ≤ (n− dr − d

)d=

(1 +

n− rr − d

)d.


Assuming for the sake of contradiction that r > n(1 − γ/d) and taking

logarithms, we get

ln1

β≤ d ln

(1 +

n− rr − d

)≤ d(n− r)

r − d<

γn

r − d<

γ

r/n− δ<

γ

1− δ − γ/d.

This can be made false by picking, e.g., γ = (1−δ) ln(1/β)1+ln(1/β)

.

2.2.3 Proof of Lemma 2.4

Restating the lemma in terms of d-evaluations, we need to show that for

every subset S ⊆ Fn2 of size R = 2r, rankd(S) ≥(r≤d

). Let S = x1, . . . , x2r

be the set of points. We simplify S by applying a sequence of transformations

that do not increase its d-rank until we arrive to the linear space Fr2×0n−r.We now define our basic non-linear transformation Π, mapping the set S

to a set Π(S) of equal size and not greater d-rank. Informally, Π tries to set

the first bit of each element in S to zero, unless this results in an element

already in S (and in this case Π keeps the element unchanged). The oper-

ator Π was used in other contexts of extremal combinatorics, and is usually

referred to as the compressing or shifting operator (see, e.g., [2, 63].)

For y = (y1, . . . , yn−1) ∈ Fn−12 , denote by 0y and 1y the elements (0, y1, . . . , yn−1)

and (1, y1, . . . , yn−1) in Fn2 , respectively. Extend this notation to sets by writ-

ing 0T = 0y : y ∈ T, 1T = 1y : y ∈ T for a set T ⊆ Fn−12 .

We define the following three sets in Fn−12 .

T∗ = y ∈ Fn−12 : 0y ∈ S and 1y ∈ S ,

T0 = y ∈ Fn−12 : 0y ∈ S and 1y /∈ S ,

T1 = y ∈ Fn−12 : 0y /∈ S and 1y ∈ S .

Writing S as

S = 0T∗ ∪ 1T∗ ∪ 0T0 ∪ 1T1 ,

we define Π(S) to be

Π(S) = 0T∗ ∪ 1T∗ ∪ 0T0 ∪ 0T1 ;

namely, we set to zero the first bit of all the elements in 1T1. It is easy to

see that |Π(S)| = |S| as Π(S) introduces no collisions.


Proposition 2.9. rankd(Π(S)) ≤ rankd(S).

Proof. It will be easier to prove this using an alternative definition for rankd(S).

Let (x1, . . . , x2r) be some ordering of S. For a degree d polynomial p ∈RM(n, d), let vp ∈ F2r

2 be the evaluation of p on the points of S

vp = (p(x1), p(x2), . . . , p(x2r)) .

Consider the linear space of vectors vp for all p ∈ RM(n, d). The dimension

of this space is exactly rankd(S), as the monomials used in the definition of

d-rank form a basis for the space of polynomials.

But now, instead of the dimension, consider the co-dimension. We call

a point xi, 1 ≤ i ≤ 2r, dependent if there are coefficients α1, . . . , αi−1 ∈ F2

such that for all degree d polynomials

p(xi) =i−1∑j=1

αjp(xj) .

We thus expressed rankd(S) as the number of independent points in S,

which is the same as the difference between |S| = 2r and the number of

dependent points in S. To prove that rankd(Π(S)) ≤ rankd(S), it suffices to

show that Π maps dependent points in S to dependent images in Π(S). Let

us consider an ordering of S in which the elements of 1T1 come last. Since all

other points in S are mapped to themselves by Π, it is clear that dependent

points in S appearing before 1T1 are also dependent in Π(S). It remains to

prove the claim for points in 1T1.

Let t1 = |T1| and let y1, . . . , yt1 be some ordering of T1. Assume 1yi ∈ S is

dependent and we will show that 0yi ∈ Π(S) is also dependent. By definition,

there exist coefficients αy, βy, γy, δy such that, for any degree d polynomial,

p(1yi) =∑y∈T∗

αyp(0y) +∑y∈T∗

βyp(1y) +∑y∈T0

γyp(0y) +∑

yj∈T1:j<i

δyjp(1yj) .

Each polynomial p ∈ RM(n, d) can be uniquely decomposed as

p(x1, . . . , xn) = x1p′(x2, . . . , xn) + p′′(x2, . . . , xn) ,


where p′ ∈ RM(n − 1, d − 1) and p′′ ∈ RM(n − 1, d). Moreover, for every

y ∈ Fn−12 , we have that p(0y) = p′′(y) and p(1y) = p′(y) + p′′(y). Since p′

and p′′ are independent, we can decompose the dependency of p(1yi) into its

p′ and p′′ components as follows.

p′(yi) =∑y∈T∗

βyp′(y) +

∑yj∈T1:j<i

δyjp′(yj) , (2.2)

p′′(yi) =∑y∈T∗

(αy + βy)p′′(y) +

∑y∈T0

γyp′′(y) +

∑yj∈T1:j<i

δyjp′′(yj) . (2.3)

We now move to consider Π(S). Every 1yi for yi ∈ T1 is mapped to 0yi,

so we should only consider the p′′ component for T1’s elements. Also, by the

definition of T∗ and T0, for each y ∈ T∗ ∪ T0, 0y ∈ S ∩ Π(S). By (2.3), for

any p ∈ RM(n, d),

p(0yi) =∑y∈T∗

(αy + βy)p(0y) +∑y∈T0

γyp(0y) +∑

yj∈T1:j<i

δyjp(0yj) ,

that is, 0yi is also dependent in Π(S).

Therefore, we have established that rankd(Π(S)) ≤ rankd(S).

We now combine our basic Π with invertible linear transformations to

define a wider class of simplifying transformations. For any u, v ∈ Fn2 whose

inner product is 〈u, v〉 = 1, we define the mapping Πu,v as follows. Informally,

Πu,v tries to add v to elements x of S for which 〈u, x〉 = 1, unless this results

in an element already in S. In other words, if both x and x + v are in S,

then Πu,v(S) maps them both to themselves. Otherwise, if just one of them

is in S, it maps it to x if 〈u, x〉 = 0, and to x + v if 〈u, x+ v〉 = 0. This is

well defined as 〈u, v〉 = 1. Note that Πe1,e1 ≡ Π.

Formally, let A be an n × n invertible matrix such that eT1A = u and

A−1e1 = v. We can construct such invertible A since 〈u, v〉 = 1 by setting

the first row of A to be u and the remaining rows of A to be a basis for the

(n− 1)-dimensional space normal to v. Define Πu,v = A−1ΠA.

Observe that invertible affine transformations do not change the d-rank

of a set, as they act as permutations on the set of degree d polynomials.

Combining this with Proposition 2.9, we get that Πu,v maintains the size

of S without increasing the d-rank.

2.3 Proof of Proposition 2.3 39

We now use a sequence of Πu,v applications to transform the set S into the

linear space V = Fr2 × 0n−r spanned by the first r unit vectors e1, . . . , er.

We say that x ∈ S is good if x ∈ V , and is bad otherwise. If all the elements

of S are good then S = V since all the elements of S are distinct. Otherwise,

let x ∈ S be some bad element and let x′ ∈ V \ S. Since x /∈ V , there must

be some index r < i ≤ n such that xi = 1; set u = ei and v = x+ x′.

We show that applying Πu,v maps x to x′ and does not affect any good

elements, thus increasing the number of good elements. First see that 〈u, v〉 =

vi = xi + x′i = 1 + 0 = 1 since x′ ∈ V so Πu,v is well defined. See also that as

〈u, x〉 = xi = 1 and x+v /∈ S, Πu,v will add v to x, transforming it to x′ ∈ V .

Also, any good element y is unchanged by Πu,v since 〈u, y〉 = yi = 0. In

total, the number of good elements increased by at least one.

We repeat this until all elements are good, that is, until S is transformed

to V , establishing that rankd(S) ≥ rankd(V ). To finish the proof, observe

that the restriction of polynomials in RM(n, d) to points in a linear space

of dimension r is exactly RM(r, d). Since |RM(r, d)| =(r≤d

)(see [93]), we

get that for any set S of size 2r,

rankd(S) ≥(

r

≤ d

),

as required.

2.3 Proof of Proposition 2.3

Let d < γn for a constant γ < 1/2. We define a set of polynomials with

measure of at least 2−c′2(

n≤d) such that all polynomials in this set have a bias

of at least 2−c′1n/d (for constants c′1, c

′2). That is, we will prove

Prf∈RM(n,d)

[bias(f) > 2−c

′1n/d]≥ 2−c

′2(

n≤d) .

Similar to the proof of Theorem 2.1, we divide the n variables into two

sets: V ′ of size n′ = dn/de and V ′′ of size n′′ = n − n′. Consider the set

of monomials of degree at most d that are multilinear in V ′ (and thus have

degree at most d− 1 in V ′′).


We first show that the number of such monomials is only a constant factor

smaller than the number of all monomials of degree at most d. The number

of monomials we consider is(n′

1

)(n′′

≤ d− 1

)≥ n

d

(n(1− 1/d)

d− 1

).

There exists a constant cγ > 0 such that if d < γn then(n(1− 1/d)

d− 1

)≥ cγ

(n

d− 1

)and also

(n

d

)≥ cγ

(n

≤ d

).

Hence the number of monomials multilinear in V ′ is at least c2γ

(n≤d

).

Let L be the linear space of polynomials on these monomials, |L| ≥2c

2γ( n≤d). Consider a random polynomial f ∈ L. Since each monomial of f

has exactly one variable in V ′, we can decompose f as the sum of products

of a variable from V ′ and a random degree d− 1 polynomial from V ′′. That

is, if V ′ = x1, . . . , xn′ and V ′′ = xn′+1, . . . , xn, we can write

f(x1, . . . , xn) =n′∑i=1

xigi(xn′+1, . . . , xn) .

We now show f has an expected bias of 2−n′ ≥ 2−n/d. Consider a partial

assignment to the variables x1, . . . , xn′ of V ′. If all of them are zero, then

f(0, . . . , 0, xn′+1, . . . , xn) ≡ 0, and hence has bias 1. In all other cases, we are

left with a random degree d− 1 polynomial in the variables from V ′′ and as

such it has bias 0 (e.g., since the constant term is random). Thus,

Ef∈L [bias(f)] = 1 · Pr [∀1 ≤ i ≤ n′ : xi = 0]

+ 0 · Pr [∃1 ≤ i ≤ n′ : xi 6= 0] = 2−n′,

and we get that

Pr[bias(f) > 2−(n′+1)

∣∣∣f ∈ L] > 2−(n′+1) .

We conclude that there is a constant c′2 such that

Pr[bias(f) > 2−(n/d+1)

]≥ Pr [f ∈ L]·Pr

[bias(f) > 2−(n/d+1)

∣∣f ∈ L] ≥ 2−c′2(

n≤d) .

Chapter 3

Polynomial threshold functions:

Structure and approximation


We study the computational power of polynomial threshold functions,

that is, threshold functions of real polynomials over the boolean cube.

We show that low-degree polynomial threshold functions cannot approx-

imate any function with many influential variables. We provide a couple of

examples where this technique yields tight approximation bounds. To this

end we prove that every polynomial threshold function admits a relatively

small partial assignment which results in a function with no influential vari-

ables.

3.1 Introduction

A boolean function h : −1, 1n → −1, 1 is a threshold (or sign) function

of a real function f : −1, 1n → R if

h(x1, . . . , xn) = sgn(f(x1, . . . , xn)).

In this chapter we study thresholds of low-degree polynomials, or Polynomial

Threshold Functions (PTFs). There is a long line of research that study the

case of linear functions, i.e., degree 1 polynomials, which are commonly called

42 Polynomial threshold functions: Structure and approximation

Linear Threshold Functions (LTFs), or halfspaces (see, e.g., [71, 34, 55] and

their references within). A key example for an LTF is the majority function

which can be defined as

Maj(x1, . . . , xn) = sgn(x1 + · · ·+ xn).

Here we study the limits of the computational power of polynomial thresh-

old functions. We show that some broad families of functions cannot be

approximated by low degree polynomial threshold functions.

A boolean function g : −1, 1n → −1, 1 is said to be ε-approximated

by degree d PTFs, if there exists a degree d PTF h(x) s.t. Prx∈U [h(x) = g(x)] ≥1− ε.

We prove that functions whose variables have high influence cannot be

approximated by low-degree PTFs, where the influence of a variable xi in g

is defined as the probability that flipping xi changes the value of g, i.e.,

Infi(g) = Prx [g(x) 6= g(x⊕ ei)],

where ei is the i-th unit vector. Define also Inf∞(f) = maxi Infi(f). We

prove the following.

Theorem 3.1. Let g : −1, 1n → −1, 1 be a boolean function, such that

Infi(g) ≥ τ for at least nα variables. Then for any degree-d polynomial

threshold function h we have

Prx [h(x) = g(x)] ≤ 1− τ

2+ η

where η = O(d/(α log n)1/8d).

We illustrate the power of Theorem 3.1 by showing two examples. The

first one shows that MODm function cannot be approximated by low degree

PTFs, while the second result shows that any low-degree polynomials over

F2 cannot be approximated by low-degree PTFs much better than the best

trivial approximation. Define the MODm function as

MODm(x1, . . . , xn) =

1

∑ni=1

xi+12≡ 0 (mod m)

−1∑n

i=1xi+1

26≡ 0 (mod m)

Note that as xi+12∈ 0, 1, this definition is essentially equivalent to the

common one. We have the following.

3.2 Preliminaries 43

Corollary 3.2. Let h : −1, 1n → −1, 1 be a degree-d polynomial thresh-

old function for

d = o(log log n/ log log log n). Then

Pr [h(x) = MODm(x)] ≤ 1− 1

m+ o(1).

This result is tight in the sense that trivially the MODm function admits

an 1− 1m

approximation by the constant −1 function (which is also a degree-0

PTF).

Corollary 3.3. Let q : Fn2 → F2 be a degree-r depending on all variables,

and let q′(x) = −1)q(x). Let h : −1, 1n → −1, 1 be a degree-d polynomial

threshold function for d = o(log log n/ log log log n). Then

Pr [h(x) = q′(x)] ≤ 1− 2−r + o(1).

This result is essentially tight, as if q is a product of r linear forms, then

the constant 1 function gives an 1− 2−r approximation of q.

3.1.1 Related Work

Bruck [48] studied polynomial threshold functions, and proved that such

functions can be computed by depth-2 polynomial sized circuits with un-

bounded fan-in linear threshold gates. Aspnes et al. [15] studied the ap-

proximation of boolean functions by some threshold functions. Namely, they

study the best possible approximation for the parity function and other sym-

metric functions by low-degree PTF, and proved that for every degree-k PTF

p, we have

Prx [p(x) 6= PARITY (x)] ≥∑b(n−k−1)/2c

i=0

(ni

)2n

,

and this bound is tight. However, their bounds for other functions are not

fully explicit and are not tight.

3.2 Preliminaries

Given a function f : −1, 1n → R, we let Var(f) = Var f(x), where x is

distributed uniformly in −1, 1n. We extend the definition of influence for


arbitrary real functions over the cube by

Inf(f) =∑i∈S

f(S)2

,

where f(x) =∑

S f(S)χS(x) is the fourier expansion of f . See [76] for

background on fourier expansion and influences.

3.2.1 Decision trees

A Decision Tree over binary variables x1, . . . , xn is a binary tree, where each

internal node v is labeled by one of the variables xv, such that the labels

along any path from the root to a leaf are distinct. Also, the two (directed)

edges that leave each node are labeled by −1 and 1. Therefore, given a path

P from the root to a leaf, for every variable x that appears along the path

we can uniquely define a value xP ∈ −1, 1 to be the label of the edge in P

that leaves the node labeled by x.

A path P from the root to a leaf ` defines a partial assignment A` by

assigning every variable that appears on x by xP . All the variables that do

not appear on P remain unassigned.

We denote the set of variables labeling the vertices in the path to ` by

var(`). We denote the set of leaves of a decision tree D by L(D).

The depth of a leaf is the length of the path from the root to it, and the

depth of a decision tree is the maximal depth of a leaf.

With a slight abuse of notation, we define a random leaf in a decision

tree to be the result of the following procedure. We start at the root, and

at each step we move to one of his children, uniformly and independently of

the other choices. When we arrive at a leaf ` we output it. Equivalently, we

choose each leaf ` with probability 2−depth(`).

Let f : −1, 1n → R be a function, D be a decision tree on x1, . . . , xnand ` be a leaf in D. We define the restriction of f to `, denoted by f |`, to be

the function obtained by f after assigning the variables x1, . . . , xn according

to A`. Namely, the domain of f |` is −1, 1[n]\var(`), and the range of f |` is

R.

Similarly, given a distribution D, define its restriction to `, D|` to be the

the distribution obtained from D conditioning on the partial assignment A`.

3.2 Preliminaries 45

We define a random function f |D by choosing a random leaf ` of D and

restricting f to `.

3.2.2 Tail estimates for polynomials

In this subsection we prove the following lemma.

Lemma 3.4. There exist constants c1, c2 > 0 such that the following holds.

Let f(x1, . . . , xn) be a polynomial of degree d such that Var[f ] = 1. For ε > 0

let α = (c1 · ε/d)d and τ = (c2 · ε/d)8d. If Inf∞(f) ≤ τ , then for every t ∈ R,

Prx [|f(x)− t| ≤ α] ≤ ε.

We need the following two lemmas.

Lemma 3.5 (Theorem 2.1 in [94]). Let f(x1, . . . , xn) be a multilinear degree

d polynomial, such that Inf∞(f) ≤ τ . Then for every t ∈ R

|Prx [f(x) ≤ t]− Prx∈N [f(x) ≤ t]| ≤ O(dτ 1/8d).

The following is an immediate corollary of Theorem 8 in Carbery and

Wright [49], which is also stated as Corollary 3.23 in [94].

Lemma 3.6. Let f(x1, . . . , xn) be a multilinear degree d polynomial such that

Var[f ] = 1. Then for every t ∈ R,

Prx∈N [|f(x)− t| ≤ α] ≤ O(dα1/d).

Proof of Lemma 3.4. Let f be a degree-d polynomial such that Inf∞(f) ≤ τ .

By Lemma 3.5 we have

Prx [|f(x)− t| ≤ α] ≤ Prx∈N [|f(x)− t| ≤ α] +O(dτ 1/8d).

By Lemma 3.6 we have

Prx∈N [|f(x)− t| ≤ α] ≤ O(dα1/d)

Combining the two results we get


Prx [|f(x)− t| ≤ α] ≤ O(d · (τ 1/8d + α1/d)).

Setting α = (c1 · ε/d)d and τ = (c2 · ε/d)8d for some absolute constants

c1, c2 > 0 we get

Prx [|f(x)− t| ≤ α] ≤ ε.

3.3 The effect of partial assignments

Our main tool is a new structural result about PTFs. Given a polynomial

threshold function p, we show that it has a small set of variables, on which

in most of their possible assignments we obtain a restricted function with no

influential variable. More precisely, the partial assignments are given by a

small depth decision tree.

Lemma 3.7. Let f : −1, 1n → R be a degree-d polynomial with Var[f(x)] =

1, and let h(x) = sgn(f(x)). For any ε, δ > 0, there exists a decision tree D

of depth at most 2ed/δ · log(1/ε), such that

Pr`∈L(D) [Inf∞(f |`) > δ] < ε

and

Pr`∈L(D) [Inf∞(h|`) > δ′] < ε

for δ′ = O(d · δ1/8d).

We defer the proof of Lemma 3.7 to Section 3.4.

Independently of the results that are presented in this chapter, Diakoniko-

las et al. [56] proved a similar result. We state their result in our terminology.

Theorem 3.8 ( [56]). Let f : −1, 1n → R be a degree-d polynomial and

let h(x) = sgn(f(x)). For any τ > 0, there exists a decision tree D of depth1τ· (d log 1

τ)O(d) such that with probability 1− τ over a random leaf ` ∈ L(D),

the function h|` is τ -close to a PTF h′ such that Inf∞(h) < τ .

3.3 The effect of partial assignments 47

We note that while the results of [56] are quantitatively stronger, the proof

of our result is simpler, and uses only elementary arguments, whilst the result

of [56] uses hypercontractive inequalities. Moreover, for our applications,

both theorems yield essentially the same bounds.

We next apply Lemma 3.7 in order to prove our main result.

Proof of Theorem 3.1. Let g : −1, 1n → −1, 1 be a boolean function for

which Infi(g) ≥ τ for at least n′ = nα variables. We will provide a lower

bound on q = Pr [g(x) 6= h(x)],

Set δ > 0 and ε > 0 to be determined later. Set m = 2ed/δ log 1/ε and

δ′ = O(d · δ1/8d). Using Lemma 3.7 we get that there exists a decision tree

D of depth at most m, such that

Pr`∈L(D) [Inf∞(h|`) > δ′] < ε.

In each path in D there are at most m variables. Thus, there exists a

variable xi for which Infi(g) ≥ τ which appears in at most m/n′-fraction of

the paths. Equivalently, a random leaf ` ∈ L(D) assigns a value to xi with

probability at most m/n′. Note also that by the triangle inequality

Prx [g|`(x) 6= g|`(x⊕ ei)] ≤Prx [g|`(x) 6= h|`(x)] + Prx [h|`(x) 6= h|`(x⊕ ei)] + Prx [h|`(x⊕ ei) 6= g|`(x⊕ ei)]

We hence get

Pr [g(x) 6= g(x⊕ ei)] ≤ E`∈L(D) [Prx [g|`(x) 6= g|`(x⊕ ei)]] +m/n′

≤ 2Pr [g(x) 6= h(x)] + E`∈L(D) [Infi(h|`)] +m/n′

≤ 2q + δ′ + ε+m/n′

On the other hand, by assumption we have Pr [g(x) 6= g(x⊕ ei)] ≥ τ .

Combining the two bounds we get that

Pr [g(x) 6= h(x)] = q ≥ 1

2(τ − ε− δ′ −m/n′)

≥ τ

2−O(ε+ dδ1/8d + 2ed/δ log(1/ε)/n′)


Setting δ = O(d/ log n′) and ε small enough (for example ε = 1/n′) gives

q = Pr [g(x) 6= h(x)] ≥ τ

2− η

for η = O( d(α logn)1/8d

).

We finish this section by proving Corollary 3.2 and Corollary 3.3.

Proof of Corollary 3.2 . It is straightforward to verify that Infi(MODm) =2m

for all i ∈ [n], the proof now follows by Theorem 3.1.

proof of Corollary 3.3. Recall that q′(x) = (−1)q(x). We will prove

Infi(q′) ≥ 21−r for all i ∈ [n] by showing that Pr [q(x) 6= q(x⊕ ei)] ≥ 21−r.

write q(x) = xiq1(x′) + q2(x′). As q1 is a non-zero polynomial of degree at

most r− 1, we have Pr [q1(x′) = 1] ≥ 21−r, and hence we have Infi(q′) ≥ 2−r

as well. Corollary 3.3 follows by Theorem 3.1

3.4 Proof of Lemma 3.7

The proof of Lemma 3.7 will be conducted in three steps. First we show that

for every low-degree polynomial there exists a partial assignment of a small

set of variables under which we get a polynomial with low influences. We

then argue that if a polynomial has low influences, then so does its threshold.

We then conclude by showing that if there is a single good assignment, then

by taking larger set of variables we get that most of the assignments are

good. The first step is accomplished by the following lemma.

Lemma 3.9. Let f : −1, 1n → R be a degree-d polynomial. For every

δ > 0 there exist a set of variables xi1 , . . . , xik and assignments for these

variables bi1 , . . . , bik ∈ −1, 1, such that

Inf∞(f |xi1=bi1 ,...,xik=bik) ≤ δ

and k ≤ ed/δ.

3.4 Proof of Lemma 3.7 49

Proof. We construct a sequence of assignments for the variables of f , assign-

ing a value to a single variable at each step, that will lead eventually to a

polynomial f |xi1=bi1 ,...,xik=bikwhose influence is bounded by δ.

Every degree-d polynomial f can be uniquely represented as

f(x) =∑

I⊂[n],|I|≤d

fI∏i∈I

xi.

For α ≥ 0 define operator Vα(f) to be

Vα(f) =∑

I⊂[n],|I|≤d

|fI |2(1 + α)|I|.

Note that V0(f) = E [f 2].

Fix a variable xi, and let f(x) = xif1(x′) + f2(x′) where

x′ = (x1, . . . , xi−1, xi+1, . . . , xn). We have f |xi=1 = f1 + f2 and f |xi=−1 =

−f1 + f2. It is easy to verify that V0(f1) = Infi(f) · V0(f).

We first claim that

12(Vα(f |xi=1) + Vα(f |xi=−1)) = Vα(f)− αVα(f1) (3.1)

Indeed, let f1(x′) =∑f1,I

∏i∈I x

′i and f2(x′) =

∑f2,I

∏i∈I x

′i. We have

Vα(f |xi=1) + Vα(f |xi=−1) = Vα(f1 + f2) + Vα(−f1 + f2) =∑I

(f1,I + f2,I)2(1 + α)|I| +

∑I

(−f1,I + f2,I)2(1 + α)|I| =

2 ·∑I

(f 21,I + f 2

2,I)(1 + α)|I| =

2 ·∑I

(f 21,I(1 + α)|I|+1 + f 2

2,I(1 + α)|I|)− 2α ·∑I

f 21,I(1 + α)|I| =

2 · (Vα(f)− αVα(f1))

This proves (3.1). In particular for α = 0 we get

12(V0(f |xi=1) + V0(f |xi=−1)) = V0(f). (3.2)


and for α > 0 we have

12(Vα(f |xi=1) + Vα(f |xi=−1)) ≤ Vα(f)− α · Infi(f) · V0(f), (3.3)

since Vα(f1) ≥ V0(f1) = Infi(f) · V0(f).

Define Sα(f) = Vα(f)V0(f)

. We next prove that

min (Sα(f |xi=1), Sα(f |xi=−1)) ≤ Sα(f)− α · Inf(f) (3.4)

By combining (3.1) and (3.2) we get

Sα(f) =Vα(f)

V0(f)=Vα(f |xi=1) + Vα(f |xi=−1)

V0(f |xi=1) + V0(f |xi=−1)+αVα(f1)

V0(f)≥ (3.5)

min

(Vα(f |xi=1)

V0(f |xi=1),Vα(f |xi=−1)

V0(f |xi=−1)

)+αV0(f1)

V0(f)= (3.6)

min (Sα(f |xi=1), Sα(f |xi=−1)) + α · Infi(f) (3.7)

Consider the polynomial f . We first bound Sα(f),

Sα(f) =Vα(f)

V0(f)=

∑I |fI |2(1 + α)|I|∑

I |fI |2≤ (1 + α)d.

Note that either Inf∞(f) ≤ δ, or there exists a variable xi1 , such that

min(Sα(f |xi1=1), Sα(f |xi1=−1)

)≤ Sα(f)− α · δ

Consider the restriction fxi1=bi1for bi1 ∈ −1, 1 minimizing Sα(fxi1=bi1

).

Either Inf∞(fxi1=bi1) ≤ δ, or otherwise we could find another variable xi2

such that

min(Sα(f |xi1=bi1 ,xi2=1), Sα(f |xi1=bi1 ,xi2=−1)

)≤ Sα(f |xi1=bi1

)− α · δ

Continuing in this fashion, since Sα ≥ 0, we must reach after at most k ≤(1+α)d

αδsteps a polynomial f |xi1=bi1 ,...,xik=bik

such that Inf∞(f |xi1=bi1 ,...,xik=bik) ≤

δ. Choosing optimally α = 1d−1

we get k ≤ e · d/δ.

We now show that if a polynomial has low influences, then so does its

threshold.


Lemma 3.10. Let f : −1, 1n → R be a degree-d polynomial such that

Inf∞(f) = δ. Let h(x) = sgn(f(x)). Then

Inf∞(h) ≤ O(d · δ1/8d).

Proof. Assume w.l.o.g Var[f ] = 1, and we will bound Infi(h) for all i =

1, . . . , n.

We first argue that if E [f 2] is large, then h has low influences. Let f(x) =

c + f0(x), where c is the free coefficient of f . We have Var[f ] = E [f 20 ] = 1

and E [f 2] = 1 + c2. The probability that h(x) = h(0) is bounded by

Pr [h(x) = h(0)] ≤ Pr [|f0(x)| ≥ c] ≤ E [f 20 ]

c2=

1

c2.

Thus for large c we get a bound on the influence of h, since

Infi(h) = Pr [h(x) 6= h(x⊕ ei)] ≤ Pr [h(x) 6= h(0)]+Pr [h(x⊕ ei) 6= h(0)] ≤ 2/c2.

In particular if c > δ−1/4 we get that Infi(h) ≤ O(δ1/2) and we are done.

We thus assume from now on that c ≤ δ−1/4.

Let f(x) = xif1(x) + f2(x), where f1, f2 do not depend on xi. By our

assumption on the influences,

Ex[f 2

1

]= Infi(f) · E

[f 2]≤ δ(1 + c2) ≤ 2δ1/2.

Set a = δ1/8 and consider the following two cases.

1. |f(x)| ≤ a

2. |f1(x)| ≥ a

If neither of these cases occur, then flipping xi does not change the sign

of f . Thus we can bound

Infi(h) ≤ Pr [|f(x)| ≤ a] + Pr [|f1(x)| ≥ a].

We first estimate the first summand. Set δ ≥ max( dc1a1/d, d

c2δ1/8d) where

c1, c2 are the constants in Lemma 3.4. We get

Pr [|f(x)| ≤ a] ≤ δ = O(d · δ1/8d).


We proceed by estimating the second summand. By Markov inequality

and get

Pr [|f1(x)| ≥ a] ≤ E [f 21 ]

a2≤ 2δ1/4.

Combining the two estimations we get that

Infi(h) ≤ O(d · δ1/8d),

as desired.

We next prove Lemma 3.7. Using Lemma 3.9 we prove the existence of

a small depth decision tree, such that for most of its leaves, the polynomial

restricted to the leaf has low influences. We use Lemma 3.10 to argue that

when this happens also the threshold function has low influences.

Proof of Lemma 3.7. We first prove the theorem for a polynomial f , and

then for a PTF h. We build a decision tree D in steps. At every step, some

of the leaves of D will be open, and some will be closed. If a leaf ` is closed

then Inf∞(f |`) ≤ δ. A leaf is open if it is not closed. Initially, our tree

consists a single vertex, the root, which is open.

Let ` be an open leaf, and consider the polynomial f |`. By Lemma 3.9,

there exist a set of variables xi1 , . . . , xik , k ≤ edδ

and an assignment to these

variables bi1 , . . . , bik ∈ −1, 1, such that

Inf∞(f |`,xi1=bi1 ,...,xik=bik) ≤ δ.

We add under a ` a subtree whose leaves correspond to all the 2k possible

assignments of xi1 , . . . , xik . Note that at least one of the leaves in the new

tree is closed, and the other leaves may be either closed or open. Therefore,

a random walk of length k that starts at ` will end at a closed leaf with

probability at least 2−k.

This process defines a treeD′ of depth at most n, as every variable appears

in every path at most once. Let D(t) be the tree obtained by truncating D′

at depth t · 2k. Namely, the depth of D(t) is t · 2k. The probability that

a random walk that start from the root will end at open leaf is at most


(1 − 2−k)t ≤ e−2−kt. Thus, setting, t = log(1/ε) · 2ed/δ will guarantee that a

random leaf in D is closed with probability at least 1− ε, as required.

We proceed by proving the second item. Let h be a PTF as stated, and

observe that by Lemma 3.10, for any leaf ` for which Inf∞(f |`) ≤ δ we have

that Inf∞(sgn(f |`)) ≤ O(dδ1/8d) = δ′. Since sgn(f |`) = sgn(f)|` = h|`, we

get

Pr`∈L(D) [Inf∞(h|`) > δ′] < ε.


Part II

Graphs

Chapter 4

Long cycles in subgraphs of

(pseudo)random directed

graphs


We study the resilience of random and pseudorandom directed graphs

with respect to the property of having long directed cycles. For every 0 < γ <

1/2 we find a constant c = c(γ) such that the following holds. Let G = (V,E)

be a (pseudo)random directed graph on n vertices and with at least a linear

number of edges, and let G′ be a subgraph of G with (1/2 + γ)|E| edges.

Then G′ contains a directed cycle of length at least (c − o(1))n. Moreover,

there is a subgraph G′′ of G with (1/2 + γ − o(1))|E| edges that does not

contain a cycle of length at least cn.

4.1 Introduction

Given a property P , a typical problem in extremal graph theory can be stated

as follows. Given a number of vertices n, what is the minimal (or maximal)

number fP(n) such that any graph on n vertices with f(n) edges possesses

P? Many examples of such problems and results can be found, e.g., in [43].

58 Long cycles in subgraphs of (pseudo)random directed graphs

Usually, the property P we consider in extremal problems is either mono-

tone increasing or monotone decreasing. A property P is monotone increasing

(respectively, decreasing) if it is preserved under edge addition (respectively,

deletion).

The resilience of a graph G with respect to a property P measures how

far the graph is from any graph H that does not have P . In particular, the

study of resilience usually focuses on monotone properties, and the following

two types of problems are studied.

Global Resilience. Given a monotone increasing property P , the global

resilience of G with respect to P is the maximal integer R such that for every

subset E0 ⊆ E(G) of |E0| = R edges, the graph G−E0 still possesses P . For

the case of a monotone decreasing property P , the global resilience of G with

respect to P is defined as the maximum number R such that the addition of

any subset of R edges to G still results in a graph G′ ∈ P .

One can also define the notion of local resilience of a graph with respect

to, say, a monotone increasing property P as the maximum number r such

that for any subgraph H ⊆ G of maximum degree r, the graph G−H is still

in P . Since in this chapter we will be concerned with properties related to

global resilience, we will not dwell on the notion of local resilience anymore.

To the best of our knowledge, Sudakov and Vu were the first [107] to

define the notion of global resilience explicitly and quantitatively and to put

it forward as a subject of independent study (it is closely related though to

the well studied notion of fault tolerance, see, e.g., [9]). However, in a sense

many well known theorems in extremal graph theory can be stated using

this terminology. For example, given a fixed graph H, the Turan number of

H, denoted by ex(H,n), is the minimum number m such that any graph on

n vertices with m edges contains a copy of H. Clearly, the study of Turan

numbers is equivalent to the study of the global resilience of the complete

graph Kn with respect to the property of having a copy of H.

Woodall [113] gave tight bounds for the number of edges in an undirected

graph that guarantees the existence of a cycle of length at least `. In our ter-

minology, he gave tight bounds on the global resilience of Kn with respect to

the property of having a cycle of length at least `. We will discuss Woodall’s

4.1 Introduction 59

result later and will also use his result. Lewin [88] studied the analogous

problem for directed graphs, and he gave tight bounds on the number of

edges required for having a directed cycle of length at least `. Many ex-

tremal results regarding the existence of cycles in directed graphs can be

found, e.g., in [36].

Recently, there has been a series of works studying the resilience of graphs

with respect to different properties. Dellamonica et al. [54] studied the local

and global resilience of long cycles in pseudorandom undirected graphs. Kriv-

elevich et al. [87] studied the resilience with respect to pancyclicity (having a

cycle of every possible length). Ben-Shimon et al. [31] studied the resilience

of several graph properties in random regular graphs. Alon and Sudakov [14]

studied the resilience of the chromatic number in random graphs. Bottcher

et al. [46] studied the local resilience of G(n, p) with respect to the property

of having an almost spanning bounded degree bipartite graph with sublinear

bandwidth. Later, answering a question from [46], Huang et. al.[73] ad-

dressed the resilience with respect to having a spanning subgraph H. Balogh

et al. [17] studied the resilience of random and pseudorandom graphs with

respect to containing a copy of a given nearly spanning tree of bounded

maximum degree.

Here we study the resilience of pseudorandom (and hence, of random)

directed graphs with respect to the property of having a long directed cycle

(namely, a simple directed cycle that covers a constant fraction of the ver-

tices). We prove asymptotically tight bounds, and thus provide the asymp-

totic value of the resilience of every graph with respect to this property, as-

suming it has some predefined pseudorandomness property. Our proof uses

a variant of the celebrated Szemeredi’s regularity lemma for sparse directed

graphs, and a short and simple technique for finding a long directed path

in pseudorandom directed graphs. Using these techniques we can reduce

our problem to the case of undirected graphs, where by applying techniques

of [54] we can give tight bounds.


4.1.1 The models

We consider here directed graphs on n vertices, where antiparallel edges are

allowed. We say that a graph D = (V,E) has density p if |E| = pn2.

Let D(n, p) be the following probability distribution on the set of n-

vertex directed graphs. Every graph in the support of D(n, p) contains n

vertices, and for every two distinct vertices x, y, there is an edge from x to

y with probability p, and independently there is an edge from y to x with

probability p. Clearly, the expected number of edges is 2p(n2

).

Once we define our random digraph model, it is usually desirable to define

a pseudorandom analog. That is, we would like to define a property such

that graphs with this property have many of the ’nice’ properties of random

graphs. Roughly speaking, we say that a directed graph is pseudorandom

if the number of edges between every two large enough sets is close to the

expected number of edges in a random directed graph with the same density.

More formally, we say that a directed graph G is (p, r)-pseudorandom if it

has edge density p and for every two disjoint sets A,B ⊆ V (G), |A| = |B|,the number of edges from A to B, denoted by eG(A,B), satisfies

|eG(A,B)− p|A||B|| ≤ r|A|√pn.

This is (up to normalization) a directed variant of the well known notion

of jumbled graphs, that was introduced by Thomason [108]. In his celebrated

work, Thomason essentially proved that a graph distributed as G(n, p) is

(p,O(1))-pseudorandom with high probability.1 On the other hand, there is

no infinite sequence of (p, o(1))-pseudorandom graphs.

The following lemma can be easily verified by combining a Chernoff type

bound with the union bound.

Lemma 4.1. For every constant c > 0 there is a constant C > 0 such that

for p ≥ Cn

, a random directed graph G ∈ D(n, p) is (p, c)-pseudorandom with

high probability.

Our results in this chapter will hold for every (p, r)-pseudorandom graph

with p ≥ Cn

for some sufficiently large constant C, and every r ≤ µ√pn for

1Here a sequence of events An, n ≥ 1 is said to occur with high probability if

limn→∞ Pr [An] = 1.

4.1 Introduction 61

some small constant µ > 0 that does not depend on C. By Lemma 4.1, a

random directed graph distributed according to D(n, p) with p ≥ Cn

has this

property with high probability.

We show here that the directed case is both similar and different from the

undirected case. In fact, since we reduce here the directed case to the global

resilience problem of the undirected case, we can use ideas from Dellamonica

et al. [54] in order to get our bounds on the resilience for directed graphs.

On the other hand, many of the techniques that were used for the undirected

case cannot be applied in the directed case. Also, the range of parameters

relevant to us is rather different, since in particular the result of Dellamonica

et al. [54] shows that the removal of any 0.99-fraction of the edges of a

(pseudo)random undirected graph still leaves a cycle of linear size. For the

directed case it is easy to see that one can always remove half of the edges

of any directed graph and get an acyclic directed graph, and hence a graph

with no cycles at all.

4.1.2 Our results

Woodall [113] studied the minimal number of edges that guarantees the ex-

istence of a long cycle. In our terminology, he studied the global resilience

of the complete graph Kn with respect to the property of having a cycle of

length at least `. He proved the following.

Theorem 4.2 (Woodall [113]). Let 3 ≤ ` ≤ n. Every graph G on n vertices

satisfying

e(G) ≥⌈n− 1

`− 2

⌉·(`− 1

2

)+

(r + 1

2

)+ 1,

where r = (n− 1) mod (`− 2), has a cycle of length at least `.

It is easy to verify that Woodall’s bound is best possible. Indeed, take a

graph formed by dn−1`−2e disjoint cliques of size ` − 2, a single smaller clique

of size r and a vertex that is connected to every other vertex in the graph.

Clearly, the length of a longest cycle in this graph is at most `− 1.

The work of Dellamonica et al. [54] can be viewed as a generalization of

Woodall’s work from the case of Kn to the case of general pseudorandom


graphs. In order to cite their result and also for future reference here the

following function is defined.

Definition 4.3. For a given 0 ≤ α < 1, define

w(α) = 1− (1− α)b(1− α)−1c.

It is easy to verify that we have w(0) = 0 and limx1w(x) = 0. The

following asymptotic version of Woodall’s result is proved in [54].

Theorem 4.4 ( [54]). Let α > 0. For every β > 0 there is n0 such that for

every graph G on n > n0 vertices satisfying

|E(G)| ≥(n

2

)·(

1− (1− w(α))(α + w(α)) + β)

has a cycle of length at least (1− α) · n.

Observe that we can partition a vertex set of size n into k = d 11−αe sets,

each of size (1 − α)n, and a remaining set of size (w(α)n, as 1 − w(α) =

k(1− α).

Dellamonica et al. proved in [54] that Theorem 4.4 can be extended

to (sparse) pseudorandom graphs; more specifically, they proved that any

subgraph G′ = (V,E ′) of a (p, r)-pseudorandom graph G = (V,E) (where

p n−1, and r is fixed) with |E ′| ≥ (1−(1−w(α))(α+w(α))+o(1))|E| edges

has a cycle of length at least (1− α) · |V |. Here we provide tight bounds on

the resilience of pseudorandom directed graphs with respect to the property

of having a long directed cycle.

Our main theorem is a directed version of their result.

Theorem 4.5. Fix 0 < γ < 12

and let G = (V,E) be a (p, r)-pseudorandom

directed graph on n vertices, where r ≤ µ√np and µ(γ) > 0 is a sufficiently

small constant that depends only on γ and n is sufficiently large. Let G′ be

a subgraph of G with at least (12

+ γ)|E| edges. Then G′ contains a directed

cycle of length at least (1− α− o(1)) · n, where α satisfies

2γ = 1− (1− w(α))(α + w(α)).

4.1 Introduction 63

Observe crucially that every directed graph G = (V,E) contains an

acyclic subgraph G′ with at least |E|/2 edges. Indeed, fix a permutation

σ : V → V , and let G1 be the subgraph with all edges xy such that

σ(x) > σ(y), and G2 be the subgraph with all edges xy such that σ(x) < σ(y).

Then both G1 and G2 are acyclic, and at least one of them contains at least

half of the edges of G.

Theorem 4.5 yields the following two immediate corollaries.

Corollary 4.6. For every γ > 0 there is a constant c1(γ) > 0 such that

the following holds. Let G be a (p, r)-pseudorandom graph on n vertices,

r ≤ µ√np where µ(γ) > 0 is some sufficiently small constant that depends

only on γ and n is sufficiently large. Let G′ be a subgraph of G with at least

(1/2 + γ)|E(G)| edges. Then G′ contains a directed cycle of length at least

c1n.

In other words, the above corollary guarantees that the deletion of less

than half of the edges of a pseudorandom digraph leaves a cycle of linear

length.

Corollary 4.7. There exists a function c2(ε) with limε→0 c2(ε) = 0 such that

the following holds. Let G be a (p, r)-pseudorandom graph on n vertices,

r ≤ µ√np where µ(γ) > 0 is some sufficiently small constant that depends

only on γ and n is sufficiently large. Let G′ be a subgraph of G with at least

(1 − ε)|E(G)| edges. Then G′ contains a directed cycle of length at least

(1− c2) · n.

Here, we prove that deleting a sufficiently small fraction of the edges of

a pseudorandom digraph leaves a cycle of length close to n.

Finally, we prove the following matching lower bound.

Proposition 4.8. Fix 0 < γ < 12

and let G be a (p, r)-pseudorandom directed

graph on n vertices, where r = O(√np) and pn → ∞. There is a subgraph

G′ with (12

+ γ)|E| edges that does not contain any directed cycle of length at

least (1− α + o(1)) · n, where α satisfies

2γ = 1− (1− w(α))(α + w(α)).


Our Tools. One of the main tools we use in this chapter is a sparse directed

variant of Szemeredi’s regularity lemma (Lemma 4.9), that was stated in [57].

This allows us to partition our graph into a constant number of regular pairs,

and essentially to reduce the problem to finding almost spanning paths in

regular pairs.

To this end, we use a simple yet powerful lemma that finds almost span-

ning paths in expanding graphs (Lemma 4.11). In our case, a regular pair

is a bipartite expander in both directions. The approach is based on ideas

from [47, 18, 27].

The rest of the chapter is organized as follows. In Section 4.2 we state the

sparse directed regularity lemma, and prove that regular pairs have an almost

spanning directed path. In Section 4.3, we reduce the resilience problem in

directed graphs to undirected graphs, and then apply ideas from [54] and

prove Theorem 4.5. In Section 4.4 we prove Proposition 4.8 and show that

our results are essentially tight.

Throughout the chapter we assume that the order of G, denoted by n, is

large enough. We do not try to optimize constants and omit floor and ceiling

signs whenever these are not crucial.

4.2 The regularity lemma and long paths in

regular pairs

4.2.1 The regularity lemma

In this section we follow [57] and state a regularity lemma for sparse directed

graphs. We first provide some notation.

Given a directed graph G = (V,E), for any pair of disjoint sets of vertices

U,W , we let EG(U,W ) be the set of edges directed from U to W , and let

eG(U,W ) = |EG(U,W )|. We say that G is (δ,D, p)-bounded if for any two

disjoint sets U,W such that |U |, |W | ≥ δ|V | we have

eG(U,W ) ≤ Dp|U ||W |.

The edge density from a set U to a set W is defined by eG(U,W )|U ||W | . We say

that two sets U and W span a bipartite directed graph of bi-density p if it has

4.2 The regularity lemma and long paths in regular pairs 65

edge density at least p in both directions. Also define the directed p-density

from U to W by

dG,p(U,W ) =eG(U,W )

p|U ||W |.

We omit the index graph G and write dp(U,W ) whenever the base graph is

clear from the context.

For 0 < δ ≤ 1, a pair (U,W ) is (δ, p)-regular in a digraph G if for every

U ′ ⊆ U and W ′ ⊆ W such that |U ′| ≥ δ|U | and |W ′| ≥ δ|W | we have both

|dG,p(U,W )− dG,p(U ′,W ′)| < δ,

and

|dG,p(W,U)− dG,p(W ′, U ′)| < δ.

A partition P = V0, V1, . . . , Vk of V is (δ, k, p)-regular if the following

properties hold.

1. |V0| ≤ δ|V |.

2. |Vi| = |Vj| for all 1 ≤ i < j ≤ k.

3. At least (1−δ)(k2

)of the pairs (Vi, Vj), 1 ≤ i < j ≤ k, are (δ, p)-regular.

We will use the following variant of Szemeredi’s regularity lemma, that

is stated in [57], and whose proof follows lines similar to the proof of the

regularity lemma for sparse undirected graphs, proved independently by Ko-

hayakawa and by Rodl (see, e.g., [83]). In [57] the lemma is stated for oriented

graphs (where no anti parallel edges are allowed), yet the result can be easily

adjusted to our case, where anti parallel edges are allowed.

Lemma 4.9 (Lemma 3 in [57]). For any real number δ > 0, any integer

k0 ≥ 1 and any real number D > 1, there exist constants η = η(δ, k0, D) and

K = K(δ, k0, D) ≥ k0 such that for any 0 < p(n) ≤ 1, any (η,D, p)-bounded

directed graph G admits a (δ, k, p)-regular partition for some k0 ≤ k ≤ K.


4.2.2 Every regular pair contains a long path

We next prove that every regular pair of positive bi-density contains an al-

most spanning path. To this end, we first show a trivial expansion property

of regular pairs, and then use this property to prove the desired result.

Claim 4.10. Let (U,W ) be a (δ, p)-regular pair for |U | = |W | with bi-density

at least 2δp, where p > 0. Then for every two sets U ′ ⊆ U and W ′ ⊆ W such

that |U ′| ≥ δ|U | and |W ′| ≥ δ|W | there is a directed edge from U ′ to W ′.

Proof. By regularity we have

eG(U ′,W ′) ≥ (dp(U,W )− δ)p|U ′||W ′| ≥ (2δ − δ)p|U ′||W ′| = δp|U ′||W ′| > 0.

The claim follows.

We next show that a bipartite directed graph with a simple expansion

property contains a long directed path. The proof follows ideas from [47, 18].

In Chapter 5 we prove and use a variant of this lemma for non-bipartite

graphs.

Lemma 4.11. Let H = (V1, V2, E), where |V1| = |V2| = t, be a directed

bipartite graph that satisfies the following property. For every two sets A ⊆V1, B ⊆ V2 of size k, there is at least one edge from A to B and there is

at least one edge from B to A. Then H contains a directed path of length

2t− 4k + 3.

Proof. Recall that DFS (Depth First Search) is a graph search algorithm

that visits all the vertices of a (directed or undirected) graph G as follows.

It maintains three sets of vertices, letting S be the set of vertices which

we have completed exploring, T be the set of unvisited vertices, and U =

V (G)\ (S∪T ), where the vertices of U are kept in a stack (a last in, first out

data structure). It is also assumed that some order σ on the vertices of G is

fixed, and the algorithm prioritizes vertices according to σ. The algorithm

starts with S = ∅, U = ∅ and T = V (G).

While there is a vertex in V (G) \ S, if U is non-empty, let v be the last

vertex that was added to U . If v has an out-neighbor u ∈ T , the algorithm

inserts u to U . If v does not have an out-neighbor in T then v is popped out

4.2 The regularity lemma and long paths in regular pairs 67

from U and is moved to S. If U is empty, the algorithm chooses an arbitrary

vertex from T and pushes it to U .

We now proceed to the proof of the lemma. We execute the DFS algo-

rithm for an arbitrary chosen order σ on the vertices of the graph H. We

let again S, T, U be three sets of vertices as defined above. At the beginning

of the algorithm, all the vertices are in T , and at each step a single vertex

either moves from T to U or from U to S. At the end of the algorithm, all

the vertices are in S.

Consider the point during the execution of the algorithm when |S| = |T |.Observe crucially that all the vertices in U form a directed path, and we have

||U ∩ V1| − |U ∩ V2|| ≤ 1. Since |U | = 2t − |S| − |T | = 2t − 2|S| is even, we

have in fact

|U ∩ V1| = |U ∩ V2| .

We get that

|S| = |S ∩ V1|+ |S ∩ V2| = |T ∩ V1|+ |T ∩ V2| = |T |,

and

|V1 \ U | = |S ∩ V1|+ |T ∩ V1| = |S ∩ V2|+ |T ∩ V2| = |V2 \ U |.

Hence, we get both

|S ∩ V2| = |T ∩ V1| ,

and

|S ∩ V1| = |T ∩ V2| .

Assume without loss of generality that

|S ∩ V1| ≥ |S|/2 ≥ |S ∩ V2|.

Then |S ∩ V1| ≥ t/2 − |U |/4 and therefore |T ∩ V2| ≥ t/2 − |U |/4. Observe

crucially that there are no edges from S to T . By the assumption of the

lemma we conclude that |S ∩ V1| = |T ∩ V2| ≤ k − 1 and therefore we get

t/2−|U |/4 ≤ k− 1 and hence |U | ≥ 2t− 4k+ 4. Thus H contains a directed

path |U | of length 2t− 4k + 3, as desired.

We therefore have the following corollary.


Corollary 4.12. Let (U,W ) be a (δ, p)-regular pair with bi-density at least

2δp and |U | = |W | = t, p > 0. Then the bipartite directed graph between U

and W contains a directed path of length (1− 2δ) · 2t+ 2 that starts at U .

Proof. By Claim 4.10, there is an edge in each direction between every two

sets of size δt in U and W . Therefore Lemma 4.11 implies the existence of

a directed path of length (1 − 2δ)2t + 3. Note that if the first vertex in the

path is from W we may remove it, thus getting a directed path of length at

least (1− 2δ)2t+ 2 that starts at U .

4.3 Proof of Theorem 4.5

In this section we prove our main result. Given a constant γ > 0, we es-

sentially want to prove that every subgraph with a (1/2 + γ)-fraction of

the edges of a pseudorandom directed graph contains a long directed cy-

cle. Let δ = δ(γ) be a constant that we will fix later, K = K(δ, 1/δ, 1 + δ)

and η = η(δ, 1/δ, 1 + δ) be the constants defined by the regularity lemma

(Lemma 4.9).

Let G = (V,E) be a (p, r)-pseudorandom directed graph with r ≤ µ√np.

For µ ≤ δ ·minη, 1/K we have

r ≤ δ√np ·minη, 1/K.

Let A,B be two sets of vertices of size ηn in G. Observe that

r√pn|A||B| ≤ δ

√np η

√pη2n3 = δη2n2p ≤ δη2n2 = δ|A||B|.

Therefore, we get that G is (η, 1 + δ, p)-bounded.

Given a subgraph G′ = (V,E ′) of G that contains (1/2 +γ)|E| edges, our

goal is to show that G′ contains a long directed cycle.

Clearly, G′ is (η, 1 + δ, p)-bounded as well, and hence we can apply the

sparse directed regularity lemma (Lemma 4.9) to G′ and get a partition of V

to clusters V0, V1, . . . , Vm, where 1/δ ≤ m ≤ K, |V0| ≤ δn, |V1| = |V2| = · · · =|Vm| = t and all but at most a δ-fraction of the pairs (Vi, Vj) are (δ, p)-regular.

Note that n(1−δ)m≤ t ≤ n

m.

4.3 Proof of Theorem 4.5 69

We next define an undirected auxiliary graphH on the clusters V1, . . . , Vm.

With a slight abuse of notation, we denote the vertices of H by V1, V2, . . . , Vm.

Two vertices Vi and Vj are connected if the pair (Vi, Vj) is (δ, p)-regular and

has bi-density at least 2δp.

Since G is (p, r)-pseudorandom and r ≤ δ·√npK≤ δ·√np

m, we get that the

p-density of every pair (Vi, Vj) in G is at least 1− δ and at most 1 + δ.

Observe that if Vi and Vj are not connected by an edge in H, one of the

following must happen.

• The pair (Vi, Vj) is not regular.

• Either |EG′(Vi, Vj)| < 2δp|Vi||Vj| or |EG′(Vj, Vi)| < 2δp|Vi||Vj| . In other

words, at least (1 − 3δ)p|Vi||Vj| of the edges from Vi to Vj or from Vjto Vi in G are not in G′.

The number of non-regular pairs is at most δ(m2

). Since in every pair with

edge bi-density less than 2δp at least (1−3δ)pt2 edges were lost when moving

from G to G′, we get that the number of pairs in G′ with edge bi-density less

than 2δp is bounded by

(1/2− γ)pn2

(1− 3δ)pt2≤ (1/2− γ)pn2

(1− 3δ)p(n(1−δ)m

)2=

1/2− γ(1− 3δ)(1− δ)2

·m2.

Observe that by our choice of parameters for the regularity lemma we

have m ≥ 1δ, and hence we have

m2 ≤ 2

(m

2

)(1 + 2/m) ≤ 2

(m

2

)(1 + 2δ).

We conclude that the fraction of non-edges in H is bounded by

δ +(1− 2γ)(1 + 2δ)

(1− 3δ)(1− δ)2.

By taking δ γ, we get that the fraction of edges in H is at least

2γ(1−o(1)), where the o(1) term depends only on δ and can be made arbitrary

small.


Let α be the minimal solution of the equation

2γ = 1− (1− w(α))(α + w(α)),

where w is the function defined in Subsection 4.1.2. By Theorem 4.4, we get

that H contains an undirected cycle of length at least (1− α− o(1))m.

We complete the proof by showing how to construct a long cycle in G′

given a long cycle in H. We start with the case that H contains a long cycle

of even length, and later show how to modify the proof in the case of odd

length.

Let Vi1 , Vi2 , . . . Vi2b be a cycle of length (1 − α + o(1))m in H. Note

that for every 1 ≤ q ≤ 2b, the pair (Viq , Viq+1) (where we identify Vi2b+1

with Vi1) is (δ, p)-regular and has edge bi-density at least 2δp. Therefore, by

Corollary 4.12, for every 1 ≤ q ≤ b, there is a directed path Pq of length at

least (1−2δ) ·2t that alternates between Vi2q−1 and Vi2q . For every 1 ≤ q ≤ b,

let PRq = Pq ∩ Vi2q and let PL

q = Pq ∩ Vi2q−1 .

Observe that for every 1 ≤ q ≤ b, by Claim 4.10 there is an edge from the

last δt vertices of PRq to the first δt vertices of PL

q+1 (here we identify PLb+1

with PL1 ). Thus we can paste every two consecutive paths together, losing

at most 2δt vertices from each path. We conclude that we can paste all the

paths together to get a directed cycle of length st least

b · (1− 2δ − δ) · 2t = (1− α− o(1))mt = (1− α− o(1))|V (G′)|,

where the o(1) term depends only on δ(γ) and can be made arbitrary small

(as α depends on γ only).

Finally, suppose there is an odd cycle of length (1 − α + o(1))m in H,

whose vertices are by Vi1 , Vi2 , . . . Vi2b+1. As in the previous case, for every

1 ≤ q ≤ b, let Pq be a path of length at least(1 − 2δ) · 2t that alternates

between Vi2q−1 and Vi2q . Let Z ′ be the first δt vertices in PL1 and Z ′′ be the

last δt vertices in PRb . Moreover, let

V ′i2b+1= v ∈ Vi2b+1

: ∀u ∈ Z ′, (u, v) /∈ E,

and

V ′′i2b+1= v ∈ Vi2b+1

: ∀u ∈ Z ′′, (v, u) /∈ E.

4.4 Lower bounds 71

By Claim 4.10, we have |V ′i2b+1|, |V ′′i2b+1

| ≤ δt, and therefore for all but 2δt

of the vertices in Vi2b+1we have an edge from the last 2δt vertices in Pb and

an edge to the first 2δt first vertices in P1. Since t > 2δt we conclude that

we can connect Pb and P1 through a vertex in Vi2b+1, thus getting a path of

length at least

(b− 1) · (1− 2δ − δ)2t = (1− α− o(1))|V (G′)|.

Theorem 4.5 follows.

4.4 Lower bounds

Let G = (V,E) be a directed graph. Recall that by fixing a permutation

σ on the vertices, we can partition the edges of G into two acyclic sets as

follows. The first set contains all directed edges xy where σ(x) > σ(y), and

the second set contains all directed edges xy where σ(y) > σ(x). Therefore,

the global resilience of every directed graph with respect to the property of

having directed cycles is at most 1/2. Here we extend this idea and show

that our main result is asymptotically tight.

Proof of Proposition 4.8. We show that there is a subgraph G′ with a

(1/2 + γ)-fraction of the edges, whose longest directed cycle is of length at

most (1− α + o(1))n. Our approach follows [54].

Recall that G is (p, r)-pseudorandom with r ≤ µ√np and pn → ∞. We

first claim that for every two disjoint sets A,B of size Ω(n), the number of

edges from A to B is p|A||B|(1 + o(1)). Indeed, let |A| = an and |B| =

bn, and suppose that a < b. If B′ is a random subset of B of size an,

then by linearity of expectation the number of edges between A and B′ is

E(A,B) · |B′|B

. Therefore, if the number of edges between A and B is smaller

than (respectively, larger than) p|A||B|(1 + o(1)) then there is a choice of a

set B′ such that A and B′ contradicts the (p, r)-pseudorandomness of G.

We partition the vertices of G into k classes V1, V2, . . . , Vk of size (1−α)n

each, and one additional class Vk+1 of size w(α)n ≤ (1 − α)n. Let G′ be

the subgraph with all edges from Vi to Vj, for 1 ≤ i < j ≤ k + 1, and all

the edges within each class Vi. Clearly, G′ does not contain a cycle longer


than (1−α)n, since a directed path leaving a certain Vi cannot return there.

Therefore, we can conclude the proof by showing that the number of edges

in G′ is as claimed.

Indeed, since G is (p, r)-pseudorandom and each Vi is of size Ω(n), we

get that for every 1 ≤ i < j ≤ k, the number of edges from Vj to Vi is

(1 − α)2 · pn2(1 + o(1)). The number of edges from Vk+1 to ∪i≤kVi in G is

k(1 − α)w(α) · pn2(1 + o(1)). Recalling that by definition of w(x) we have

(1− α)k = 1−w(α), we get that the number of edges we deleted from G to

get G′ is

((k

2

)(1− α)2 + k(1− α)w(α)

)· pn2(1 + o(1))

= (1− α)k((1− α)(k − 1) + 2w(α))(1 + o(1))pn2

2

= (1− w(α))(α− w(α) + 2w(α)) · pn2(1 + o(1))

2

= (1− w(α))(α + w(α)) + o(1))pn2

2.

Note that the number of edges in G is pn2(1 + o(1)). Let the number of

edges in G′ be (1/2 + γ) · pn2. Then the number of edges we deleted satsifies

(1 + o(1))(1/2− γ)pn2 = (1− w(α))(α + w(α)) + o(1))pn2

2,

and therefore γ satisfies

2γ = 1− (1− w(α))(α + w(α)) + o(1)),

as claimed. The statement follows.

4.5 Concluding remarks

We studied the global resilience of pseudorandom directed graphs with re-

spect to the property of having a long directed cycle. We gave matching

lower and upper bounds, and our proof essentially reduced our problem to

the case of undirected graphs.

4.5 Concluding remarks 73

A variety of questions regarding the resilience of directed graphs can be

asked. A few, somewhat arbitrary examples are the problem of local resilience

with respect to having a long directed cycle, the resilience with respect to the

property of having some fixed directed graph. Another interesting problem

is the resilience with respect to Hamilitonicity, which in the dense case is

settled in [72].

In this chapter we considered subgraphs with a (1/2 + γ)-fraction of the

edges, and observed that every directed graph contains an acyclic subgraph

with a 1/2-fraction of the edges. In the next chapter we will prove that every

two-coloring of the edges of a pseudorandom digraph contains a relatively

long monochromatic path. That is, instead of proving that a large subgraph

has a certain property, it is proved that every partition of the edges of the

graph has a certain property. It will be interesting to give such results for

other properties of directed graphs.


Chapter 5

The size Ramsey number of a

directed path


Given a graph H, the size Ramsey number re(H, q) is the minimal num-

ber m for which there is a graph G with m edges such that every q-coloring

of E(G) contains a monochromatic copy of H. We study the size Ramsey

number of the directed path of length n in oriented graphs, where no antipar-

allel edges are allowed. We give nearly tight bounds for every fixed number

of colors, showing that for every q ≥ 1 there are constants c1 = c1(q), c2 such

that

c1(q)n2q(log n)1/q

(log log n)(q+2)/q≤ re(

−→Pn, q + 1) ≤ c2n

2q(log n)2.

Our results show that the path size Ramsey number in oriented graphs is

asymptotically larger than the path size Ramsey number in general directed

graphs. Moreover, the size Ramsey number of a directed path is polynomially

dependent in the number of colors, as opposed to the undirected case.

Our approach also gives tight bounds on re(−→Pn, q) for general directed

graphs with q ≥ 3, extending previous results.

76 The size Ramsey number of a directed path

5.1 Introduction

Given an integer q > 0, we write G → (H)q if every q-coloring of E(G)

contains a monochromatic copy of H.

The study of size Ramsey numbers (initiated in [59]) is concerned with

the following questions. Given a graph H, what is the minimum number of

edges m for which there is a graph G with m edges such that G→ (H)q?

Denote by re(H, q) the size Ramsey number of H with respect to coloring

with q colors. That is,

re(H, q) = min|E(G)| | G→ (H)q.

The study of re(Kn, q) is essentially equivalent to the study of the original

Ramsey number. Namely, it can be verified that if re(Kn, q) = m then a

clique with exactly m edges has the desired property. This result is attributed

to Chvatal in [59].

Beck proved [22] that for every constant q > 0, the size Ramsey number

of the undirected path on n vertices is linear in n, answering a question of

Erdos. Namely, there is a constant cq such that

re(Pn, q) ≤ cqn.

Later Alon and Chung [10] provided an explicit construction of graphs with

this property. The linearity of the size Ramsey number for bounded degree

trees was proved by Friedman and Pippenger [64].

Directed graphs. In this chapter we focus on the size Ramsey number in

directed graphs, and in particular on the size Ramsey number of a directed

path on n vertices.

Raynaud [98] proved that every red-blue coloring of the complete sym-

metric directed graph (a directed graph in which between every two vertices

there are edges in both directions) has a Hamilton cycle which is the union

of two monochromatic paths (a simple proof can be found in [70]). In par-

ticular, this shows that the size Ramsey number of the path−→Pn for directed

graphs with antiparallel (opposite) edges is O(n2). Reimer [100] proved that

every digraph with the property that every red-blue coloring has a path of

5.1 Introduction 77

length n must have Ω(n2) edges, therefore proving that Raynaud’s bound for

non-simple directed graphs is tight up to a constant factor.

In this chapter we provide nearly tight bounds for the size Ramsey number

of directed paths in oriented graphs, where no antiparallel edges are allowed.

We show that this number is asymptotically larger than the path size Ramsey

number when such edges are allowed. Our approach also generalizes to the

case of non-simple directed graphs, and we provide nearly tight bounds for

such graphs for every constant q ≥ 3.

Our first result is a lower bound on the size Ramsey number.

Theorem 5.1. For every q ≥ 1 there is a constant c1 = c1(q) such that

re(−→Pn, q + 1) ≥ c1n

2q(log n)1/q

(log log n)(q+2)/q.

We also have the following almost matching upper bound.

Theorem 5.2. There is an absolute constant c2 such that for every q ≥ 1

re(−→Pn, q + 1) ≤ c2n

2q · (log n)2.

Moreover, a random tournament TN on N = Θ(nq log n) vertices satisfies

TN → (−→Pn)q+1 with high probability.

We stress that Theorem 5.1 proves that the size Ramsey number of a

directed path in oriented graphs is asymptotically larger than the size Ramsey

number in general directed graphs for any fixed number of colors.

In the course of the proof of Theorem 5.2, we actually prove the following

asymmetric Ramsey property.

Theorem 5.3. There is an absolute constant c such that the following holds.

There is an oriented graph G with cn2 · (log n)2 edges such that every red-blue

coloring of E(G) contains a red path of length n or a blue path of length

n log n.

Our approach also gives matching lower and upper bounds for the case of

directed non-simple graphs with more than 2 colors, where antiparallel edges

are allowed. This generalizes the results of Raynaud [98] and Reimer [100]

to any fixed q.


Proposition 5.4. The size Ramsey number of a directed path on n vertices

for q + 1 colors in directed, non-simple graphs is Ω(n2q).

Proposition 5.5. The size Ramsey number of a directed path on n vertices

for q + 1 colors in directed, non-simple graphs is O(n2q).

Note that here we show that the size Ramsey number (both for simple or

non-simple directed graphs) of a directed path is polynomially dependent in

the number of colors, in contrast to the undirected case where changing the

number of colors changes the size Ramsey number by a constant factor, as

follows from the above mentioned result of Beck [22].

Organization of the chapter. The rest of the chapter is organized as

follows. In Section 5.2 we provide some notation and definitions and present

well known theorems that will be used later on. In Section 5.3 we present

the proofs of our lower bounds, proving Theorem 5.1 and Proposition 5.4. In

Section 5.3 we prove the upper bounds, showing Theorem 5.2, Theorem 5.3

and Proposition 5.5. Finally, in Section 5.5 we present some concluding

remarks and open problems.

Throughout the chapter we assume that the underlying parameter n is

large enough. We do not try to optimize constants, and all logarithms are in

base 2. We ignore all floor and ceiling signs whenever these are not crucial.

5.2 Preliminaries

All the graphs we consider are directed. An oriented (or simple) graph is

a graph with no antiparallel (or opposite) edges. Given a set of vertices

A ⊆ V , we denote by E(A) the set of edges in the induced subgraph G[A].

The in-degree of a vertex v, denoted by d−(v), is the number of edges that

are directed into v, and the out-degree of v, denoted by d+(v) is the number

of edges directed from v. The degree of v is d+(v) + d−(v), and we let ∆(G)

be the maximum degree in a graph G and δ(G) be the minimum degree in

G. For a vertex v we let N+(v) = u ∈ V : (v, u) ∈ E and call this set

the out-neighbors of v. We also let N−(v) = u ∈ V : (u, v) ∈ E and

call this set the in-neighbors of v. For a set of vertices A we let N+(A) be

5.3 Lower bounds 79

⋃a∈AN

+(a). A set of vertices is acyclic if it does not span a directed cycle.

The length of a directed path is the number of edges it contains. The edge

density of a directed graph G = (V,E) is |E||V |2 .

The complete graph on n vertices, denoted by Kn, is an undirected graph

for which every pair of vertices are connected. A complete symmetric directed

graph is a non-simple directed graph where between every two vertices there

are edges in both directions. A tournament is an oriented graph where be-

tween every two vertices there is an edge in exactly one of the directions.

A k-coloring of a graph is a mapping of the vertices to 1, . . . , k such that

every two adjacent vertices are mapped to distinct values. The chromatic

number of a graph G, denoted by χ(G), is the minimum k such that G

is k-colorable. It is well known that every graph of maximum degree d is

(d + 1)-colorable. A Hamilton cycle in G is a cycle that visits every vertex

in G exactly once.

We will use the following two theorems several times throughout the chap-

ter. The first one, also mentioned in the introduction, is due to Raynaud [98]

(see, e.g., [70] for a proof).

Theorem 5.6. Let G be a complete symmetric directed graph on t vertices.

Then every red-blue coloring of E(G) contains a Hamilton cycle which is the

union of two monochromatic paths. In particular, it contains a monochro-

matic path of length t/2.

We also need the following simple theorem that was proved independently

by Gallai [65] and Roy [101] (see, e.g., [89], Chapter 9, Problem 9, for a proof).

Theorem 5.7. Let G be a directed graph with no path longer than t. Then

G is (t+ 1)-colorable.

5.3 Lower bounds

In this section we show that every oriented graph with relatively few edges

has an edge coloring without a long monochromatic path. We first prove

that every sparse oriented graph has a large acyclic set, and then use it to

show that every graph admits a partition into a relatively small number of


independent sets and acyclic sets. We conclude by showing that given such

a partition we can color the edges with no long monochromatic path.

5.3.1 Sparse graphs have large acyclic sets

The following lemma is well known and easy.

Lemma 5.8. Every tournament on n vertices has an acyclic set of size

blog nc+ 1.

Proof. We may assume that n = 2k for some integer k, as otherwise we can

take any subotournament of size 2blognc. We prove it by induction on k, and

note that the case k = 0 is trivial. Suppose that the induction hypothesis

is true for k, and we next prove it for k + 1. Indeed, every tournament on

2k+1 vertices contains a vertex v with out-degree at least 2k. By induction

hypothesis, N+(v) contains an acyclic set A of size k + 1, and thus A ∪ vis an acyclic set of size k + 2, as required.

Since every oriented graph is a subgraph of a tournament, we get the

following direct consequence.

Corollary 5.9. Every oriented graph on n vertices has an acyclic set of size

log n.

Our main result in this subsection is that every sparse oriented graph

has a larger acyclic set. This is a directed version of a lemma by Erdos and

Szemeredi [62]. Although it can be derived from their result, we include here

a self contained proof for the sake of completeness.

Lemma 5.10. There is an absolute constant c > 0 such that the following

holds. Every oriented graph G with n vertices and at most εn2 edges, where

ε ≥ 1n

, contains an acyclic set of size c lognε log (1/ε)

.

Proof. Let G be an oriented graph with n vertices and εn2 edges. We can

assume that ε < 1/4 as otherwise the lemma follows easily by taking c small

enough and applying Corollary 5.9. Let G′ = (V,E) be the subgraph of G

obtained by removing every vertex with in-degree greater than 2εn. Observe

that we removed at most n/2 vertices from G and therefore |V | ≥ n/2.

5.3 Lower bounds 81

Let U be a maximum acyclic set in G′, and assume that |U | < logn10ε log (1/ε)

.

Since ε ≥ 1/n, we can also assume that |U | < n/20 as otherwise we are done.

Since every vertex has in-degree at most 2εn, we have∑v∈U

d−(v) < 2εn|U |.

Let R∗ be the set of vertices in V \ U with at least 5ε|U | out-neighbors

in U , then |R∗| ≤ 2n5

, as otherwise the total number of edges directed into U

is more than 2εn|U |. Let R = V \ (U ∪ R∗) be the set of vertices outside U

with at most 5ε|U | out-neighbors in U , and we conclude that

|R| ≥ n/2− 2n/5− |U | ≥ n/20.

For every vertex v ∈ R we define a set of vertices Sv ⊆ U of size exactly

5ε|U | that contains N+(v)∩U . Using the inequality(nk

)≤ ( en

k)k, we get that

the total possible number of subsets of U of this size is(|U |

5ε|U |

)≤( e

5ε

)5ε|U |≤ 2

logn2 log (1/ε)

·log e5ε ≤ n1/2.

Therefore, by the pigeonhole principle, there is a set R′ ⊂ R of size at leastn

20n1/2 = n1/2

20such that |N+(R′) ∩ U | ≤ 5ε|U |. Also, by Corollary 5.9, R′

contains an acyclic set R′′ ⊆ R′ of size at least 12· (log n − 10). Note that

R′′ ∪ (U \N+(R′)) is an acyclic set of size at least

|R′′|+ |U | − 5ε|U | ≥ |U |+ 1

2

(log n− 10

)− log n

2 log (1/ε)> |U |,

assuming that ε < 1/4 and n is large enough. Therefore, we get a contradic-

tion as U is not a maximum acyclic set, and the lemma follows.

5.3.2 Acyclic colorings and coloring acyclic sets

In this subsection we give two building blocks that will be used later in the

proof of Theorem 5.1. We first consider the case where we are given a coloring

of the edges in some induced disjoint sets without a long monochromatic

path, and we wish to color the edges between them while keeping the length


of a longest monochromatic path relatively small. We then consider the case

when we are given an acyclic set, and show how to color its edges with no

long monochromatic path.

Given a set of vertices A and a coloring ϕ of E(A), let `ϕ(A) be the length

of a longest monochromatic path of in A with respect to ϕ.

We prove the following.

Lemma 5.11. Let A1, . . . , Ak be disjoint sets of vertices in a directed graph,

and let ϕ be a coloring of⋃ki=1E(Ai) with (q + 1) colors such that for every

1 ≤ i ≤ k we have `ϕ(Ai) ≤ r. Then ϕ can be extended to a (q + 1) coloring

ϕ′ of E(⋃ki=1Ai) such that

`ϕ′

(k⋃i=1

Ai

)≤ q(r + 1) · d q

√ke.

Proof. Let s be the minimal integer such that k ≤ sq. For each 1 ≤ i ≤ k,

we denote by (i) the representation of i in base-s with exactly q digits. That

is, we represent each index as a vector of length q, letting (i)y be the y’th

coordinate of i, and for each 1 ≤ y ≤ q, 0 ≤ (i)y ≤ s− 1.

We now define an acyclic coloring of the edges between the sets. For two

sets Ai and Aj, we color all the edges from Ai to Aj as follows. If there is an

index y such that (i)y < (j)y, we color all the edges in color y (if more than

one choice of y is possible, choose one arbitrarily). Otherwise, we color all

the edges from Ai to Aj in color q + 1.

Note that by the definition of the new coloring, if a monochromatic path

leaves a set Ai it will never return to this set. Also, if P is a monochromatic

path colored by 1 ≤ y ≤ q, then the y’th coordinates of all the sets visited by

the path are all distinct, and therefore there are at most s sets in the path.

For a monochromatic path P that is colored by (q+ 1), if Aj is visited by P

after it visits Ai then (i)y ≥ (j)y for every coordinate 1 ≤ y ≤ q, and there

is a strict inequality in at least one coordinate. Therefore, the sums of the

coordinates of all the sets visited by the path are all distinct, and hence such

a path visits at most sq distinct sets.

5.3 Lower bounds 83

We conclude that every monochromatic path colored by 1 ≤ y ≤ q visits

at most s ≤ sq distinct sets, and every monochromatic path colored by q+ 1

visits at most sq sets.

A longest monochromatic path contains at most r edges from each Ai,

plus one edge that leaves Ai. It visits at most qs = qd q√ke sets. Therefore,

the length of this path is bounded by q(r + 1) · d q√ke, as claimed.

In particular, we get the following immediate corollary, that generalizes

an argument of Reimer [100].

Corollary 5.12. Let G be a k-colorable graph. There is a (q + 1)-coloring

of E(G) with no monochromatic path longer than qd q√ke.

We next prove similarly that one can color the edges of an acyclic graph

Z using q colors with no monochromatic path longer than d q√te, where t is

the length of a longest directed path in Z.

Lemma 5.13. Let Z be an acyclic graph in which a longest directed path has

t edges. There is a q-coloring of E(Z) with no monochromatic path longer

than d q√t+ 1e.

Proof. Let s be the minimal number such that t+ 1 ≤ sq, and we prove that

there a q-coloring with no monochromatic path longer than s. For a vertex

v let `Z(v) be the length of a longest path in Z that ends in v, and for each

0 ≤ i ≤ t let

Zi = v : `Z(v) = i.

Note that these sets are well defined since Z is an acyclic graph. Moreover,

each Zi is an independent set and all the edges in the graph are directed from

Zi to Zj for j > i. Let (i) be the encoding of the number i in base s with

exactly q digits. Observe that for each j > i there is an index 1 ≤ y ≤ q

for which (j)y > (i)y. We now define the coloring as follows. For j > i, we

color all the edges from Zi to Zj in color y where y is an index such that

(j)y > (i)y. If there is more than one feasible choice of y, we choose one of

them arbitrarily.

Again, we get that in every monochromatic path of color y, all the sets

Zi that are visited by the path have distinct y’th coordinate, and there is at


most one vertex from each Zi. We conclude that every monochromatic path

contains at most s vertices, as required.

We will also need the following well known and simple claim, and we give

its proof for completeness.

Claim 5.14. Let G be an undirected graph with m edges. Then χ(G) ≤ 2√m.

Proof. Every optimal vertex coloring contains an edge between every two

color classes, hence m ≥(χ(G)

2

)and we get that χ(G) ≤ 2

√m as required.

Proof of Theorem 5.1. LetG = (V,E) be a directed graph with c1n2q(logn)1/q

(log logn)(q+2)/q

edges. Define

X = v ∈ V : d+(v) + d−(v) ≤(n

2q

)q.

Note that ∆(G[X]) ≤ ( n2q

)q (when G[X] is considered here as an undirected

graph), and therefore, χ(G[X]) ≤ ( n2q

)q + 1. By Corollary 5.12, there is a

(q + 1)-coloring of E(X) with no monochromatic path longer than n/2 + q.

Let Y = V \ X, and |Y | = m. Since |E(G)| ≤ c1n2q(logn)1/q

(log logn)(q+2)/q and every

vertex in Y has at least ( n2q

)q incident edges, we have by double counting

m ≤ 2|E|( n

2q)q≤ 2c1(2q)qnq(log n)1/q

(log log n)(q+2)/q. (5.1)

Consider the following procedure that partitions most of the vertices in

Y into Θ(log log n) families Y (1), Y (2), . . ., where each family is composed of

not too many acyclic sets.

Roughly speaking, we partition our vertices into acyclic sets as follows.

We maintain an index i, and at the i’th step we find acyclic sets that cover

approximately m/2i vertices, and group these sets to a family Y (i). To this

end, at the beginning of each step we fix a number ai for which we are

guaranteed by Lemma 5.10 that throughout the i’th step we can always find

an acyclic set of size ai. By the end of the procedure, most of the vertices

are partitioned into families, and each family is composed of acyclic sets of

the same size.

5.3 Lower bounds 85

We initiate the procedure by taking i, j = 1 and Y ′ = Y , where i repre-

sents the step number. At the beginning of step i, we fix εi = |E(Y ′)|(m/2i)2

and

ai = c log (m/2i)εi log (1/εi)

(where c is an absolute constant given by Lemma 5.10), and

also set j := 1. The step ends when |Y ′| ≤ m/2i.

At the i’th step, we repeatedly find an acyclic set Zi,j in Y ′ of size ai, and

set Y ′ := Y ′ \ Zi,j and then increase j by one.

The procedure terminates when |E(Y ′)| ≤ n2q

(16q)2q.

We first stress that for every i, j we can find an acyclic set Zi,j as claimed.

Observe that during each step, the value εi is fixed, the number of edges does

not increase, and the number of vertices is at leastm/2i. Hence, εi is an upper

bound on edge density during the step. Therefore, Lemma 5.10 guarantees

the existence of an acyclic set Zi,j of size exactly ai.

Let Y (i) = Zi,j be the family of acyclic sets that is constructed at step

i, and let ki = |Y (i)|. We next bound the size of a longest monochromatic

path in⋃kij=1 Zi,j for every i, and start with i = 1.

Let ε be the edge density in Y . Then ε = ε1/4. It is easy to verify thatlogm−1

log (1/ε)−2≥ logm

log (1/ε)(since clearly m2 ≥ 1/ε) and hence the size of each acyclic

set is at least

c log (m/2)

ε1 log (1/ε1)=

c(logm− 1)

4ε(log (1/ε)− 2)≥ c logm

4ε log (1/ε).

As we complete the first step just after we cover at least m/2 vertices

with acyclic sets, we get that the number of acyclic sets satisfies k1 ≤(m/2)·4ε log (1/ε)

c logm+ 1. Moreover, since all acyclic sets are of the same size, the

size of each Z1,j is bounded by m/2k1−1

.

By Lemma 5.13, we get that the edges of each acyclic set E(Z1,j), for

every 1 ≤ j ≤ k1, can be colored with (q + 1) colors with no monochromatic

path longer than q+1

√m/2k1−1

+ 1. We then can apply Lemma 5.11 to color the

edges between Z1,j1 and Z1,j2 for each j1, j2. The number of sets is k1, and

the size of a longest monochromatic path in each of them is bounded byq+1

√m/2k1−1

+ 1. We conclude that the edges of⋃Z1,j admit a (q + 1)-coloring


with no monochromatic path longer than

q( q√k1 + 1) ·

q+1

√m/2

k1 − 1+ 2

≤ 2q

(mqk1

2q

) 1q(q+1)

.

Note that the edge density ε in Y satisfies

ε ≤ |E|m2

=c1n

2q(log n)1/q

m2(log log n)(q+2)/q. (5.2)

Moreover, since we assume that the number of edges is at least n2q

(16q)2q

(as otherwise the procedure terminates before this step begins), we get that1ε

= m2

|E(Y )| ≤m2(16q)2q

n2q and therefore by (5.1)

log (1/ε) ≤ log

((16q)2q

n2q· 4c2

1(2q)2qn2q(log n)2/q

(log log n)(2q+4)/q

)≤ 2 log log n, (5.3)

assuming that n is large enough.

Therefore, we have the following bound.

mqk1

2q≤ mq+1ε log (1/ε)

2q−2c logm

≤ mq−1c1n2q(log n)1/q log log n

2q−4c logm · (log log n)(q+2)/q

≤ 8(2q)q(q−1)cq1nq(q+1) log n log log n

(log log n)q+2c logm

≤ 8(2q)q(q−1)cq1nq(q+1)

c(log log n)q+1

Where the first inequality follows from the bound on k1, the second one

from substituting ε and 1/ε and using (5.2) and (5.3), and the third one one

from substituting m (and applying (5.1)). The last inequality follows from

the bound log n ≤ logm that clearly holds.

Taking c1 <c1/q

8(2q)q ·(16q2)q+1 (we do not try to optimize the dependence in

q here), we get that

2q

(mqk1

2q

) 1q(q+1)

≤ 1

8q· n

q√

log log n.

5.3 Lower bounds 87

It is not difficult to verify that taking smaller values of m only decreases

the last expression. Hence, by repeating the same coloring method, we get

that for each i, there is a (q + 1)-coloring of the edges spanned by the union

of the sets in Y (i) with no monochromatic path longer than 18q· nq√log logn

. Note

that the number of vertices in Y ′ is decreased by a factor of at least 2 in each

step. Moreover, the procedure terminates when |E(Y ′)| ≤ n2q

(16q)2q, and thus

it essentially terminates if |Y ′| ≤ nq

(16q)q. Hence by applying (5.1) we get that

the number of families is bounded by

log(16q)qm

nq≤ log

((16q)q

nq· 2c1(2q)qnq(log n)1/q

(log log n)(q+2)/q

)≤ log log n,

assuming again that n is large enough.

Let W =⋃i,j Zi,j the set of vertices that were covered throughout the

procedure. We color all the edges in E(W) whose endpoints belong to sets

from distinct Y (i)’s with (q + 1) colors according to Lemma 5.11. Since the

length of each monochromatic path spanned by the sets of each family is

bounded by 18q· nq√log logn

, and the number of families is bounded by log log n,

we get that the induced subgraph of all vertices that are contained in sets

from⋃Y (i) admits a (q + 1)-coloring with no path longer than

q ·(

1

8q· n

q√

log log n+ 1

)·(q√

log log n+ 1)≤ n/4.

Finally, after the procedure terminates, we are left with a set Y ′ that

satisfies |E(Y ′)| ≤ n2q

(16q)2q. By Claim 5.14, we get that G[Y ′] is 2nq

(16q)q-colorable.

Hence Corollary 5.12 implies that E(Y ′) admits a (q + 1)-coloring with no

monochromatic path longer than n/8 + 1.

We therefore found a partition of V (G) into three sets X, Y ′,W . The

set E(X) admits a (q+ 1)-coloring with no monochromatic path longer than

n/2 + q. The set E(Y ′) admits a coloring with no monochromatic path

longer than n/8, and E(W) admits a coloring with no monochromatic path

longer than n/4. We color all the edges that are either from X to Y ′ or

from X to W or from Y ′ to W by the first color, and all the edges that are

either from Y ′ to X or from W to X or from W to Y ′ by the second color.

Every monochromatic path in E(G) that leaves one of these three sets does


not return to this set. Therefore, the length of a longest monochromatic

path is bounded by the sum of the lengths of longest monochromatic paths

in X, Y ′,W , plus at most two edges between them. Thus E(G) admits a

coloring with no monochromatic path longer than n/2+q+n/8+1+n/4+2 <

n, and Theorem 5.1 follows.

Proof of Proposition 5.4. Let G be a non-simple directed graph with(n3q

)2qedges. By Claim 5.14, G is 2

(n3q

)q-colorable, and hence by Corol-

lary 5.12 it admits a (q + 1)-coloring with no monochromatic path longer

than n, and the proposition follows.

5.4 Upper Bounds

In this section we provide an oriented graph for which every q-coloring of its

edges contains a long monochromatic path. We start with the case of two

colors. In Subsection 5.4.1 we define the notion of a k-pseudorandom digraph,

and show that a random tournament on n vertices is Θ(log n)-pseudorandom

with high probability. We next show that every red-blue coloring of a k-

pseudorandom digraph on n vertices contains a directed red path of length

Ω(nk) or a directed blue path of length Ω(n). This proves Theorem 5.3. We

conclude this section by showing how to reduce the case of any fixed number

of colors to the case of two colors, proving Theorem 5.2 and Proposition 5.5.

For notational convenience we use n in this section to denote the number of

vertices in a Ramsey digraph G , rather than the length of a target path Pnas in Theorem 5.2, Theorem 5.3 and Proposition 5.5.

5.4.1 Pseudorandom digraphs

We start with the following natural definition for k-pseudorandomness of

directed graphs.

Definition 5.15. We say that a directed graph G is k-pseudorandom if for

every two disjoint sets A,B such that |A|, |B| ≥ k, there is at least one edge

of G from A to B.

5.4 Upper Bounds 89

We first show the existence of a Θ(log n)-pseudorandom digraph by show-

ing that a random tournament satisfies this property with high probability.

Claim 5.16. A random tournament on n vertices is 2 log n-pseudorandom

with high probability.

Proof. Let Tn be a random tournament and fix two disjoint sets A,B of size k

in Tn. Since every edge is oriented in each way uniformly and independently

of the other edges, the probability that all the edges are directed from B to

A is exactly 2−k2. Since clearly (k!)2 ≥ 22k−2 there are at most

(nk

)2 ≤ n2k

(k!)2≤

22k·logn−2k+2 choices of ordered pairs of sets of size k, and thus by the union

bound the probability that there are two sets of size k for which all the edges

are oriented in one direction is bounded by

22k·logn−k2−2k+2 = o(1),

for k = 2 log n.

We also need the following property of k-pseudorandom digraphs.

Claim 5.17. Let G be a k-pseudorandom directed graph, and let A1, A2, . . . , Atbe disjoint sets, each of size at least 2k. Then there is a directed path

v1v2 . . . vt, where for each 1 ≤ i ≤ t, vi ∈ Ai.

Proof. We say that a vertex uj ∈ Aj is good if there is a directed path

ujuj+1 . . . ut such that us ∈ As, s = j, . . . , t. Clearly, our goal is to prove

that there is a good vertex in A1. Denote by A∗j the set of good vertices in

Aj. By definition, every vertex in At is good, and thus |A∗t | ≥ 2k. Also, if

uj+1 ∈ A∗j+1 and there is an edge from uj to uj+1, then uj ∈ A∗j . Using a

reverse induction, assume that we know that for some j ≤ t, |A∗j | ≥ k, then

since G is k-pseudorandom all but at most k of the vertices in Aj−1 have an

edge to A∗j . Therefore, all but at most k of the vertices in Aj−1 are actually

in A∗j−1, and thus |A∗j−1| ≥ k. We conclude that |A∗1| ≥ k and thus there is a

path as required.

The following lemma shows that every k-pseudorandom graph contains

a long path. This is a non-bipartite version of Lemma 4.11 from Chapter 4.

The proof follows ideas from [47, 18].


Lemma 5.18. Let G be a k-pseudorandom oriented graph on n vertices.

Then G contains a directed path of length n− 2k + 1.

Proof. Recall that the DFS (Depth First Search) is a graph algorithm that

visits all the vertices of a (directed or undirected) graph G as follows. It

maintains three sets of vertices, letting S be the set of vertices which we

have completed exploring them, T be the set of unvisited vertices, and U =

V (G) \ (S ∪ T ), where the vertices of U are kept in a stack (a last in, first

out data structure). The DFS starts with S = U = ∅ and T = V (G).

While there is a vertex in V (G) \ S, if U is non-empty, let v be the last

vertex that was added to U . If v has a neighbor u ∈ T , the algorithm inserts

u to U and repeats this step. If v does not have a neighbor in T then v is

popped out from U and is inserted to S. If U is empty, the algorithm chooses

an arbitrary vertex from T and pushes it to U .

We are now proceed to the proof of the lemma. We Execute the DFS

algorithm. We let again S, T, U be three sets of vertices as defined above.

At the beginning of the algorithm, all the vertices are in T , and at each step

a single vertex either moves from T to U or from U to S. At the end of

the algorithm, all the vertices are in S. Therefore, at some point we have

|S| = |T |. Observe crucially that all the vertices in U form a directed path,

and that there are no edges from S to T . We conclude that |S| ≤ k− 1, and

therefore |U | ≥ n− 2k + 2, so there is a directed path with n− 2k + 1 edges

in U , as required.

5.4.2 The case of two colors

In this subsection we prove that every k-pseudorandom directed graph on n

vertices has a monochromatic red path of length Ω(nk) or a monochromatic

blue path of length Ω(n), and this will prove Theorem 5.3. We prove the

following main lemma.

Lemma 5.19. Let G be a k-pseudorandom directed graph on n vertices.

Then every red-blue coloring of its edges yields a red directed path of lengthn

28kor a blue directed path of length n/28.

Proof. Fix a red-blue coloring of E(G). Let GR be the red graph, that con-

tains only the red edges.

5.4 Upper Bounds 91

If GR contains a directed path of length n14k

, we are done. Otherwise,

by the Gallai-Roy theorem (Theorem 5.7), GR is n14k

-colorable and therefore

has a partition into n14k

independent sets. We partition these independent

sets into sets of size exactly 7k, rounding down the remaining vertices (in

particular, we remove every independent set smaller than 7k). Since we

remove at most 7k · n14k

= n/2 vertices, we remain with t ≥ n14k

independent

sets B1, B2, . . . Bt, each containing exactly 7k vertices.

Note that each Bi, 1 ≤ i ≤ t, spans a k-pseudorandom graph and contains

only blue edges. Therefore, by Lemma 5.18, each Bi contains a blue path of

length at least 5k. Since there is a directed edge from the last k vertices of

this path to the first k vertices in the path, we conclude that each Bi contains

a directed blue cycle Ci of length at least 3k.

We next construct an auxiliary complete symmetric directed auxiliary

graph H on t vertices in which each vertex corresponds to a cycle Ci, and with

a slight abuse of notation we denote these vertices by C1, C2, . . . , Ct. We color

the edge from Ci to Cj by blue if there are at least k vertices in Ci that have

blue edges directed towards Cj, and red otherwise. Since H is complete and

symmetric, by Raynaud’s theorem (Theorem 5.6), it has a monochromatic

path of length t/2 ≥ n28k

. Let Ci1 , Ci2 , . . . , Cit/2 be the vertices in H along

the path, each such vertex represents a cycle. We complete the proof by

considering the following two cases.

H contains a red path of length t/2. For each Cij there is a set Rij for

which only red edges are going towards Cij+1. Moreover, for every 1 ≤ j ≤ t/2

we have |Rij | ≥ 2k. Observe that all the edges from Rij to Rij+1are red, and

therefore by Claim 5.17, there is a red path of length t/2 ≥ n28k

with exactly

one vertex from each Rij , as required.

H contains a blue path of length t/2. Call a vertex in Cij an endpoint

if it has a blue edge towards Cij+1. By the assumption, there are at least k

endpoints in Cij for every 1 ≤ j ≤ t/2, and therefore from each vertex in Cijthere is a path of length at least k− 1 that ends at some endpoint, in which

we can travel along one additional edge towards Cij+1. We construct a blue

path of length n28

by taking an arbitrary path of length k − 1 that ends at


some endpoint in Ci1 , moving through the endpoint to Ci2 , walking through

such a path to an endpoint of Ci2 and so on till we arrive at Cit/2 , where

we can again walk along a path of length at least 3k − 1 (that visits all the

vertices in Cit/2). We conclude that there is a blue path of length at least

k × n28k

= n28

, as claimed.

Lemma 5.19 and Theorem 5.3 follow.

Explicit constructions. Given an explicit construction of a k-pseudorandom

tournament on n vertices, our approach shows that every red-blue coloring

of such a tournament has a monochromatic path of length Ω(nk). A simple

construction of a k-pseudorandom tournament is given by Quadratic Residue

tournaments, defined as follows (see [13], Chapter 9). Let p ≡ 3 mod 4 be

a prime number. The vertices of the tournament Tp are all the elements in

the finite field Zp. For two vertices i and j, there is an edge from i to j if

and only if i − j is a quadratic residue. It can be shown that since p ≡ 3

mod 4, −1 is a quadratic nonresidue and therefore for each i, j either there

is an edge from i to j or an edge from j to i but not both. This construction

gives k = Θ(√n) (see [3] and [13]).

A much better construction can be based on explicit constructions of

Ramsey type bipartite graphs that are provided in [19, 20]. In particular,

Barak et al. [20] provided an explicit construction of an N by N zero-one

matrix such that every K by K sub-matrix is neither an all-ones matrix

nor an all-zeros matrix, where K = 2logo(1)N . Observe that this implies the

existence of a 2K-pseudorandom tournament on N vertices. Indeed, given

such a matrix M , define a tournament T = ([n], E) as follows. For every

1 ≤ i < j ≤ n, direct the edge from i to j if Mi,j = 1 and direct it from j to

i if Mi,j = 0. Note that T depends only on the entries above the diagonal of

M . Consider two disjoint sets of vertices A,B in T of size 2K. Without loss

of generality we can assume that there is a set A′ ⊂ A and a set B′ ⊆ B, each

of size K, such that all the indices of vertices in A′ precede all the indices

of vertices in B′. Therefore, the set of edges between A′ and B′ corresponds

to some K by K sub-matrix above the diagonal, which is not a constant

matrix. We conclude that there is at least one edge from A′ to B′, and at

5.4 Upper Bounds 93

least one edge from B′ to A′. This provides an explicit construction of a

2logo(1)N -pseudorandom tournament on N vertices.

5.4.3 The general case

Here we prove by induction on q that for every k-pseudorandom directed

graph G on 28knq vertices we have G → (−→Pn)q+1. Theorem 5.2 will clearly

follow by the fact that a random tournament is Θ(log n)-pseudorandom with

high probability (Claim 5.16).

The base case of the induction (q = 1) follows directly from Lemma 5.19.

Suppose that the result holds for q colors and we next prove it for q+1 colors.

Indeed, let G be a k-pseudorandom graph on 28knq vertices. Fix a coloring of

E(G) with the colors 1, 2, . . . , q+ 1. Denote by Gq+1 ⊆ G the subgraph with

all edges that are colored q + 1. If Gq+1 contains a monochromatic path of

length n we are done. Otherwise, by the Gallai-Roy Theorem (Theorem 5.7),

we get that Gq+1 is n-colorable and hence contains an independent set A of

size 28knq

n= 28knq−1.

Note that A spans a subgraph of G and hence G[A] is k-pseudorandom.

Also, the colors of all the edges of G[A] are among 1, 2, . . . , q. Therefore,

by the induction hypothesis G[A]→ (−→Pn)q, and we conclude that G contains

a monochromatic path of length n, as desired.

Proof of Proposition 5.5. The proof for non-simple directed graphs fol-

lows similar lines. Note that the Gallai-Roy theorem is valid for non-simple

directed graphs as well. Therefore, we can use the same induction on q. The

base case (q = 1) follows from Raynaud’s theorem (Theorem 5.6). Suppose

that the result holds for q colors, the correctness for q + 1 colors follows by

taking a complete symmetric graph on nq vertices, and considering the sub-

graph with all edges colored by the (q + 1)’th color. If this graph contains

a directed path of length n we are done, otherwise we find an induced sub-

graph of order nq−1 in which all edges are colored 1, 2, . . . , q, and applying

the induction hypothesis, Proposition 5.5 follows.


5.5 Concluding remarks and open problems

We proved nearly tight bounds for the size Ramsey number of a directed path

for oriented graphs. We proved that every red-blue coloring of the edges of a

k-pseudorandom graph on n vertices contains a red path of length Ω(nk) or a

blue path of length Ω(n), but it might be the case that this approach can also

give better symmetric Ramsey bounds. An interesting question is whether

every red-blue coloring of a k-pseudorandom graph contains a monochromatic

path of length Ω( n√k). Clearly every progress in this direction will improve

our upper bounds.

Another related question is about the asymptotic behavior of the maxi-

mum length of a monochromatic path in every red-blue coloring of a random

tournament. Here we proved that every tournament T has a coloring with no

monochromatic path longer than O( n√logn

), and also that with high probabil-

ity a random tournament Tn has a monochromatic path of length Ω( nlogn

) in

every red-blue coloring. It would be very interesting to close the gap between

these bounds.

When proving the lower bound on the size Ramsey number, we study

the minimal number k for which a certain graph can be partitioned into k

acyclic sets. This parameter was studied, e.g., in [8], and it was conjectured

that in every oriented graph G = (V,E) there is an acyclic set of size (1 +

o(1)) |V |2

|E| · log |E||V | . If this conjecture is true, then our lower bound can be

slightly improved.

It is easy to verify that there is no k-pseudorandom oriented graph on n

vertices for k ≤ logn2

, as every such graph has an acyclic set of size log n and

therefore has two sets of size logn2

with no edges in one of the directions. On

the other hand we proved that for k = 2 log n such graphs do exist. Hence,

it will be interesting to determine the minimum k for which there is a k-

pseudorandom oriented graph on n vertices. Another appealing question is

to provide better explicit constructions of k-pseudorandom oriented graphs.

Chapter 6

Biased orientation games


We study biased orientation games, in which the board is the complete

graph Kn, and Maker and Breaker take turns in directing previously undi-

rected edges of Kn. At the end of the game, the obtained graph is a tourna-

ment. Maker wins if the tournament has some property P and Breaker wins

otherwise.

We provide bounds on the bias that is required for a Maker’s win and

for a Breaker’s win in three different games. In the first game Maker wins

if the obtained tournament has a cycle. The second game is Hamiltonicity,

where Maker wins if the obtained tournament contains a Hamilton cycle.

Finally, we consider the H-creation game, where Maker wins if the obtained

tournament has a copy of some fixed graph H.

6.1 Introduction

In this chapter we study orientation games. The board consists of the edges of

the complete graph Kn. In the (p : q) game the two players, called Maker and

Breaker, take turns in orienting (or directing) previously undirected edges.

Maker starts the game and at each round, Maker directs at most p edges and

then Breaker directs at most q edges (usually, we consider the case where

p = 1 and q is large). The game ends where all the edges are oriented, and

96 Biased orientation games

then we obtain a tournament. Maker then wins if the tournament has some

fixed property P , and Breaker wins otherwise. Here we focus on the 1 : b

game, which is referred to as the b-biased game. We stress that at each

round, each player has to orient at least one edge, so the number of rounds is

clearly bounded. Also, Maker (respectively, Breaker) can orient up to p edges

(respectively, up to q edges) and hence by bias monotonicity every property

P admits some threshold t(n,P) so that Maker wins the b-biased game if

b < t(n,P) and Breaker wins the b-biased game if b ≥ t(n,P)

This game is a variant of the well studied classical Maker-Breaker game,

which is defined by a hypergraph (X,F) and bias (p : q). In that game, at

each round Maker claims p elements of X, and Breaker claims q elements of

X. Maker wins if by the end of the game he claimed all the elements of some

hyperedge A ∈ F , and Breaker wins otherwise. Usually, a typical problem

goes as follows. Given a game hypergraph H = (X,F), |X| = n, determine

or estimate the threshold function tH such that if b > tH then Maker wins in

a (1 : tH) game, and if b ≤ tH then Breaker wins in a (1 : tH) game. There

has been a long line of research that studies the bias threshold of various

games (see, e.g., [23, 24, 51, 66, 84, 86] and their references).

Here we study orientation games for the following three properties.

Creating a cycle. Maker wins if the obtained tournament contains a cycle,

and Breaker wins otherwise. It is well known that a tournament contains a

cycle if and only if it contains a directed triangle (cycle of length 3). This is a

relatively old question which has already been studied by Alon (unpublished

result) and by Bollobas and Szabo [44], and here we improve their results.

Creating a Hamilton cycle. Here Maker wins if the obtained tournament

contains a Hamilton cycle, and Breaker wins otherwise. Recently, the second

author [84] solved a long standing question and provided tight bounds on the

bias threshold for Maker win in the classical Maker-Breaker Hamiltonicity

game. We use a variant of his approach, together with a new application of

the Gebauer-Szabo method [66] and give tight bounds in our case as well.

6.1 Introduction 97

Creating a copy of H. Here we are given a fixed graph H. Maker wins if

the obtained tournament contains a copy of H, and Breaker wins otherwise.

We provide both upper and lower bounds, and give some nearly tight bounds

for specific cases. We conjecture that the correct threshold should be a

function of the size of the minimal feedback arc set of H, and provide some

results that support this conjecture.

Our results. In this chapter we study the cycle game, the Hamiltonicty

game and the H-creation game. We stress that in all these games Maker

wins if the obtained tournament has the desired property, no matter who

directs each edge. Our first theorem considers the cycle creating game. It

is easy to observe that if b ≥ n − 2 then Breaker has a winning strategy

(for completeness, we give a detailed proof in Section 6.3). Bollobas and

Szabo [44] proved that if b = (2−√

3)n, Maker wins the game and conjectured

that the correct threshold is b = n− 2.

We provide a simple argument that improves their result. We have the

following.

Theorem 6.1 (The cycle game). For every b ≤ n/2−2, Maker has a strategy

to create a cycle in the b-biased orientation game.

The second game we consider is the Hamiltonicity game, where Maker

wins if and only if the obtained tournament contains a Hamilton cycle. Here

we apply old and recent techniques [51, 66, 84] to get tight bounds on the

bias threshold for a win of Breaker.

Theorem 6.2 (The Hamiltonicity game).

1. If b ≥ n(1+o(1))lnn

, Breaker has a strategy to guarantee that in the b-biased

orientation game the obtained tournament has a vertex of in-degree 0,

and in particular to win the hamiltonicity game.

2. If b ≤ n(1+o(1))lnn

, Maker has a strategy to create a Hamilton cycle in the

b-biased orientation game.

In the H-creation game we have some partial results. We conjecture that

the bias that guarantees Maker’s win depends on the minimal feedback arc


set of H, and support this result for graphs with a small feedback arc set.

We will give and discuss corresponding notion in Section 6.5.

6.2 Preliminaries

Let Kn be the complete graph on n vertices, a tournament is an orientation

of Kn. A directed graph is called oriented if it contains nor loops neither

cycles of length 2. Every oriented graph is a subgraph of a tournament. A

directed graph is strongly connected if for every two vertices u, v there is a

directed path from u to v and a directed path from v to u. All directed

graphs we consider here are oriented, i.e., do not have parallel or opposite

edges.

All logarithms are in base 2 unless stated otherwise.

The classical Maker and Breaker game goes as follows. Given a hyper-

graph H = (V,F), at every round Maker occupies p elements from V , and

then Breaker occupies q elements from V . By the end of the game, Maker

wins if he occupies completely some hyperedge in F , and otherwise Breaker

wins. The well known results of Erdos and Selfridge [61] and Beck [21] give

a sufficient condition for a Maker’s win.

Theorem 6.3. Suppose that Maker and Breaker play a (p : q)-game on a

hypergraph H = (V,F). If∑A∈F

(q + 1)−|A|p <

1

q + 1,

Then Breaker has a winning strategy, even if Maker starts the game.

An orientation game is defined by a series of moves by Maker and Breaker.

In every round, Maker orients 1 ≤ mt ≤ p edges (usually in our settings

p = 1) and Breaker orients 1 ≤ bt ≤ q edges (usually in our settings q = ω(1)).

The game ends where all the edges are oriented, so the obtained graph is a

tournament. Maker wins if the tournament has some predetermined property

P , otherwise Breaker wins.

We denote by Ht the obtained oriented graph after t rounds. Clearly, this

graph has at most (p+ q) · t edges.

6.3 The cycle game 99

Given a directed graph G = (V,E), we write (u, v) ∈ E if there is an edge

from u to v. Given a set A ⊆ V , we let

N+(A) = u ∈ V \ A : ∃v ∈ A, (v, u) ∈ E,

and

N−(A) = u ∈ V \ A : ∃v ∈ A, (u, v) ∈ E.

A tournament T on n vertices is transitive if there is a bijection σ :

V (T ) → [n] such that for every edge (u, v) ∈ E(T ), σ(u) < σ(v). A tour-

nament T = (V,E) is k-colorable if there is a partition of V into k sets

V1, . . . , Vk such that the induced tournament on each Vi is transitive. Thus,

a transitive tournament is 1-colorable.

6.3 The cycle game

In this section we prove Theorem 6.1. Namely, we show that in the (n/2−2)-

biased game Maker can create a cycle. For the sake of completeness we

also prove that in the (n − 2)-biased game Breaker can create an acyclic

tournament.

Breaker’s strategy. Suppose that b ≥ n − 2, we show that Breaker can

block all cycles in the graph as follows. Whenever Maker orients an edge

from u to v, Breaker responds by orienting all edges from u to every vertex

w ∈ V (Kn) such that the edge uw has not been oriented yet. Clearly, Breaker

in his turn has to orient at most n− 2 edges.

We proceed by proving that no cycle is created when Breaker applies this

strategy. Indeed, suppose that a cycle C is created and let (u, v) the first

edge in C that was oriented (by either Maker or Breaker), and suppose also

that (w, u) ∈ C. If Maker orients the edge from u to v, by the strategy above

Breaker responses by orienting the edge from u to w, and thus (w, u) /∈ C. If

Breaker orients the edge from u to v, he did it because Maker oriented some

other edge from u to some vertex z. In this case, again Breaker will also

orient the edge from u to w, and therefore again (w, u) /∈ C. We conclude

that no cycle is created.


Maker’s strategy. Our main lemma states that Maker has a strategy so

Ht contains a directed path of length t throughout the game.

Lemma 6.4. In the b-biased game, Maker has a strategy SM such that for

every t ≤ n−1, the graph Ht obtained after t rounds contains a directed path

of length t.

Proof. We prove by induction that assuming that there are no cycles in the

graph, Maker can extend a longest path by one, no matter how Breaker plays.

Clearly Maker can create a path of length 1 at the first round. Suppose that

the longest path in E(Ht) is Pt = u1, u2, . . . , ur, where r ≥ t. Let v be a

vertex not in the path. Let k be the maximal index such that there is no

edge from v to uk. This is well defined as if there is an edge from v to u1

then v, u1, . . . , ur is a longer path, contradicting the maximality of Pt.

Observe first that if there is an edge in the opposite direction from uk to

v then u1, . . . , ur is not a maximal path. Indeed, if k = r then u1, . . . , ur, v

is a longer path; Otherwise by the definition of k there is an edge from v

to uk+1 and therefore u1, . . . , uk, v, uk+1, . . . , r is a longer path, and in both

cases this contradicts the maximality of Pt.

Therefore Maker in his turn orients the edge from uk to v and creates a

path of length at least r + 1, and the result follows.

The strategy of Maker is as follows. At each round, if he can close a

cycle he does so and wins. Otherwise, he increases the length of a longest

directed path. We next show that after large enough number of rounds,

Breaker cannot block all possible cycles.

Proof of Theorem 6.1. As long as Maker cannot orient an edge such

that a cycle is created, Maker can extend a longest directed path by 1 by

Lemma 6.4. After t rounds, there is a path Pt of length at least t. Let

Vt = V (Pt). There are(t2

)− t potential edges in G[Vt] such that orienting

any of them creates a cycle.

Consider the graph Ht−1 just before Maker starts round t. There are(t−1

2

)−(t−1) edges that may close a cycle, of them at most (b+1)(t−1)−(t−1)

were oriented in previous rounds. If (b+1)(t−1)− (t−1) <(t−1

2

)− (t−1) at

the beginning of round t then Maker wins. Unless Maker wins before that,

6.4 The Hamiltonicity game 101

the game lasts at least(n2)b+1

rounds, and therefore by taking t ≥ (n2)b+1

we get

that if b ≤ n/2− 2 then Maker surely wins.

6.4 The Hamiltonicity game

In this game Maker wins if the obtained tournament contains a Hamilton

cycle, and Breaker wins otherwise. We start with the following easy and well

known lemma, whose proof is given here for completeness.

Lemma 6.5. Let T be a strongly connected tournament. Then T contains a

Hamilton cycle.

Proof. Let C = u1, u2, . . . ur, u1 be a longest directed cycle in T . If C is not

a Hamilton cycle, there is a vertex v /∈ C. Since T is strongly connected,

there is a path from v to C and a path from C to v. Suppose first that

(ui, v), (v, uj) ∈ E(T ) for some 1 ≤ i 6= j ≤ r. Without loss of generality,

assume that j > i. Since T is a tournament, there is some index i ≤ k ≤j − 1 such that (uk, v), (v, uk+1) ∈ E(T ) and hence we get a longer cycle

u1, u2, . . . , uk, v, uk+1, . . . , ur, u1, a contradiction.

If there are no two indices i, j such that (ui, v), (v, uj) ∈ E(T ), then all

the edges between v and C are in the same direction. Suppose that for every

1 ≤ i ≤ r, we have (ui, v) ∈ E(T ) (the other case is similar). Since T

is strongly connected, there is a path v, x1, . . . , xt, ui for some 1 ≤ i ≤ r,

where the vertices x1 . . . , xt are not in C. We therefore get a longer cycle

u1, . . . , ui−1, v, x1, . . . , xt, ui, . . . , ur, u1, a contradiction. Therefore T contains

a Hamilton cycle, as claimed.

We conclude that if Maker constructs a strongly connected graph from

his own edges then he wins the game.

Breaker’s strategy. Assuming that the bias is sufficiently large, Breaker

has a strategy to guarantee that the obtained tournament T contains a vertex

with in-degree 0. In this case clearly T does not contain a Hamilton cycle.

To this end, we reduce this problem to a box game, similarly to the treatment

in [51].


Let Kn be the complete graph on n vertices, and consider the b-biased

game, where b ≥ (1+o(1))nlnn

. Recall that Ht is the oriented graph obtained after

t rounds. Fix a partition V (Kn) = A ∪ B, where A and B are disjoint sets,

|A| = b, |B| = n − b. Throughout the game, Breaker orients the edges from

A to B until after some round t there are two vertices u, u′ ∈ A such that

for every vertex w ∈ B, both (u,w), (u′, w) ∈ Ht, and the in-degree of both

u, u′ is 0. Then in the last turn he orients edges within A so that either u or

u′ will have in-degree 0.

We have the following well-known result of Chvatal and Erdos [51].

Theorem 6.6. Suppose that there are r disjoint sets (or boxes) B1, . . . , Br,

each box Bi containing k elements. At each round, Box-Maker claims b

elements and then Box-Breaker claims a single element. If

k ≤ br∑i=1

1

i,

then Maker has a strategy to occupy all the elements of a single box.

Note that in each round, Box-Breaker destroys a single box, and so

throughout the game Box-Maker tries to claim all elements of a single box

before it is destroyed by Box-Breaker.

Here we need a variant of this theorem, for the case that Box-Maker

actually has to complete two boxes.

Claim 6.7. Suppose that there are r disjoint sets, B1, . . . Br, each Bi con-

taining k elements. At each round, Box-Breaker destroys one set and then

Box-Maker claims b elements. If

k + b ≤ bk∑i=1

1

i,

then Box-Maker has a strategy to occupy all the elements of two boxes.

Proof. For every box Bi we add a set B′i of b virtual items. Consider a

standard box game where the i’th box is Bi ∪ B′i, and suppose that Box-

Maker always claim the elements of Bi before he claims the elements of B′i,


for every 1 ≤ i ≤ r. If

k + b ≤ br∑i=1

1

i,

then by Theorem 6.6 Box-Maker has a strategy to win the game. Consider

the last round before Box-Maker wins, when the next move should be taken

by Box-Breaker. Since Box-Breaker cannot avoid Box-Maker’s win there are

at least two indices i 6= j such that all but at most b elements of boxes i and

j are already claimed by Box-Maker. Therefore we conclude that there are

at least two indices i 6= j such that Bi and Bj are claimed. We conclude that

Box-Maker claimed all the elements of two of the original boxes, no matter

what Box-Breaker did. The claim follows.

In our setting, Maker and Breaker switch their roles. That is, we define

the boxes so that Breaker will take Box-Maker’s role, and if he claims a box

the obtained tournament has a vertex of in-degree 0. For every vertex v ∈ Awe define a box Xv as vw : w ∈ B. Note that |Xv| = |B| = n− b. In every

turn, Maker (that is, Box-Breaker) can destroy one box Xv by directing an

edge towards v, either from a vertex from A or from B. On the other hand,

Breaker (Box-Maker) can orient b edges from A to B, which is equivalent to

taking b elements from the various boxes. By Claim 6.7, if

n = |Xv|+ b ≤ b

|A|∑i=1

1

i, (6.1)

then Breaker has a strategy to have two vertices u, u′ from A for which all

their incident edges that connect them to B are directed towards B, and

none of the edges from A enters u or u′. Therefore, no matter what Maker

does, Breaker can direct all the edges from either u or u′, thus creating a

vertex with in-degree 0 and destroying any chance for creating a Hamilton

cycle. Taking b ≥ n(1+o(1))lnn

satisfies (6.1) and thus Breaker wins the game,

and thus Item 1 in Theorem 6.2 follows.

Maker’s strategy. Maker’s strategy consists of two stages. His goal in the

first stage is to create a graph with some expansion properties, so that all


sufficiently small sets have at least one in-going edge and at least one out-

going edge. To this end, he will create a graph with min in-degree and out-

degree at least 3. We will show that with positive probability (and actually,

with high probability) after this stage the graph has the desired expansion

properties. We conclude that Maker has a deterministic strategy that

guarantees these properties after the first stage. Moreover, the first stage

lasts at most 8n rounds in any case.

At the second stage, Maker will ensure that for every large enough disjoint

sets of vertices A,B there is at least one edge from A to B and at least one

edge from B to A. We will show that if the he succeeds at the first stage then

after the second stage we will have a strongly connected graph and hence by

the end of the game Maker will win.

We say that a directed graph G is k-expanding if the following holds.

• For every set A of size at most k, |N+(A)|, |N−(A)| > 0.

• For every two disjoint sets A,B of size at least k, there is an edge from

A to B and there is an edge from B to A.

We will show that after the first stage the obtained graph will have the

first property with high probability, and after the second stage it will have

the second property.

We have the following.

Lemma 6.8. Let G be a directed graph, and suppose that G is k-expanding

for some k. Then G is strongly connected.

Proof. Let A1, A2, . . . , At be the strongly connected components of G, and

suppose that t > 1. Let T be a graph where each Ai is represented by a

vertex, and there is an edge from Ai to Aj if and only if there is a vertex

vi ∈ Ai and a vertex vj ∈ Aj such that (vi, vj) ∈ E(G). It is well known that

T is a directed forest, and therefore contains a leaf, i.e., a set Ai with no

outgoing edges. If |Ai| < k then since |N+(Ai)| > 0 we get a contradiction.

If |Ai| > n− k, then since |N−(V \Ai)| > 0 we get a contradiction. Finally,

if k ≤ |Ai| ≤ n − k, then by the second property there is an edge from

Ai to V \ Ai. Therefore, we conclude that t = 1 and hence G is strongly

connected.


More specifically, we will show that for k = n(lnn)2/5

, at the first stage

Maker ensures that for every set A of size at least k, |N+(A)|, |N−(A)| > 0,

and at the second stage Maker ensures that for every two sets A,B of size at

least k, there is an edge from A to B. By Lemma 6.8 and Lemma 6.5, after

the second stage Maker wins.

The first stage. At the first stage we adapt the techniques of Gebauer and

Szabo [66] in a way similar to [84] and show that if b = (1−o(1))nlnn

then Maker

has a winning strategy. We start by reducing our game to an undirected

game on the edges of a bipartite graph.

Suppose that Maker and Breaker play a biased orientation game on

the edges of the complete graph G = (V,E) on n vertices, and let V =

v1, v2, . . . vn. Let H = (V1, V2, E′) be the complete bipartite graph on

2n vertices, where V1 = v1,1, v1,2, . . . v1,n and V2 = v2,1, v2,2, . . . , v2,n.Throughout the game we maintain two subgraphs, HM consisting of edges

that are associated with Maker and HB consisting of edges that are associated

with Breaker. Initially both graphs are empty.

If Breaker orients a previously undirected edge from vi to vj in G, we add

the edge between v2,i and v1,j to HB.

Maker, in his turn, would like to create a graph with a constant minimum

degree in HM . Whenever Maker, according to the strategy to be described

below, wants to add some edge (v1,i, v2,j) to HM and (v2,i, v1,j) has not been

taken yet, he does it and also orients vi to vj in G (note that in this case the

edge between vi and vj is undirected before this step). In this case we also

add the edge (v2,i, v1,j) to HB. If, on the other hand, (v2,i, v1,j) ∈ E(HB),

then he adds (v1,i, v2,j) to HM , and then plays another turn by taking a free

edge according to his strategy, and adding the opposite edge to Breaker’s

graph. Finally, if Maker takes an edge (v1,i, v2,i then he plays another turn.

Since the classical Maker-Breaker game is bias-monotone, if Maker takes

more than one edge it can only help him. Also, note that edges from v1,i

to v2,i are useless for Maker in the real game. Therefore Maker will have to

construct a graph with minimum degree c + 1 so that every vertex has at

least c neighbors other than himself.

Observe also that (v2,i, v1,j) /∈ E(HM), as otherwise (v1,i, v2,j) would be


added to HB in some previous step. The following proposition summarizes

this reduction.

Proposition 6.9. If at some step HM has minimum degree c+ 1 then at the

same time every vertex in G has min in-degree and out-degree at least c.

Gebauer-Szabo proof. In [66], Gebauer and Szabo provided a strategy

for Maker (in the classical Maker-Breaker setting) to construct a spanning

tree, a graph with postive minimum degree, and a connected graph with

high minimum degree when b = (1+o(1))nlnn

. Here we summarize their method

and highlight the slight differences between their strategy for the min-degree

game and what we need in our case. We refer the reader to [66] for a complete

proof. Their strategy is defined as follows. The goal of Maker is to construct

a graph with min-degree c. Throughout the game, a vertex v is dangerous if

dM(v) ≤ c− 1. Define the danger value of v as dang(v) = dB(v)− 2b · dM(v).

Initially, the danger of all vertices is 0. At every round, Maker takes a vertex

v with maximum danger value (ties are broken arbitrarily), and then takes

an arbitrary unclaimed edge incident to v. The proof goes by assuming

a Breaker’s win, and analyzing the change of danger value of the vertices

for which Maker took incident edges in the game, and showing that the

average danger value must be greater than 0. This in turn would lead to a

contradiction.

In our case, our board consists of the edges of the complete bipartite graph

Kn,n instead of the edges of the complete graph Kn. Moreover, when Maker

claims an edge, Breaker may get the opposite edge as well; We add this edge

to the next move of Breaker. Therefore, Maker plays against Breaker that

claims at most (b + 1) edges in his turn. The danger of a vertex is defined

only with respect to edges (and degrees) that belong to the bipartite graph

Kn,n, and hence at the beginning of the game the danger of every vertex is

0. The rest of the analysis is essentially the same as [66].

It was observed in [84] that Maker can achieve the minimum degree c

at every vertex before Breaker claimed (1 − δ)n of its incident edges, for

δ = 15(lnn)1/4

. Also, if Maker claims an edge that is incident to a vertex v, he

chooses one of the edges randomly and uniformly among the free incident

edges. Note this in this case Breaker also gets only one new edge.


We will show that after the first stage, the obtained graph has typically

some expanding properties. In our case after at most 8n rounds, HM has

min-degree at least 4, which results in an oriented graph with the property

that every vertex has in-degree and out-degree at least 3. Observe that this

stage lasts at most 8n moves as in every round Maker increases the degree

of one of the vertices in Kn,n by at least one.

We conclude the description of this approach with the following proposi-

tion.

Proposition 6.10. Suppose that b = (1−o(1))·nlnn

. Then Maker has a strategy

to construct after at most 8n turns a directed graph with min in-degree and

min out-degree at least 4. Moreover, throughout the game, Maker chooses at

each turn a vertex v according to his strategy, and picks a random incident

edge out of a set of at least δn choices, where δ = 15(lnn)1/4

.

Applying Gebauer-Szabo approach. Let A be a set of vertices of size

O( n(lnn)2/5

). We next prove that almost surely after the first stage A has at

least one ingoing edge and at least one outgoing edge. We start by claiming

that almost surely every such set has at least one ingoing edge. Observe first

that the property trivially holds for every set with a single vertex, as every

vertex has in-degree at least one. Consider a fixed set A of size i, and assume

that A has no ingoing edges, then all edges that enter A have their other

endpoint also in A, and there are at least 3i such edges. By Proposition 6.10,

whenever Maker chooses a dangerous vertex v from A, there are at least δn

unclaimed edges incident to v. Therefore, the probability that Maker chooses

an edge between v and another vertex of A is at most |A|−1δn−1

. After the first

stage there are 3i ingoing edges to vertices of A, hence the probability that A

does not have even a single ingoing edge from a vertex outside A is at most

( |A|−1δn−1

)3i. Therefore, by the union bound, the probability that there is set A

of size i with no ingoing edge is at most(n

i

)·(|A| − 1

δn− 1

)3i

≤(eni

)i·(

2i

δn

)3i

≤(

8ei2

δ3n2

)i.

By considering the two cases when i ≤ n1/3 and i ≥ n1/3 it is easy to

check that for every 2 ≤ i ≤ n(lnn)2/5

and δ = 15(lnn)1/4

, the last expression is


bounded by o(1/n). Therefore by the union bound every set of size at mostn

(lnn)2/5has at least one ingoing edge, assuming that n is sufficiently large.

Essentially the same argument shows that almost surely every such set of

that size contains at least one outgoing edge, as claimed.

Clearly, the first stage takes at most 8n rounds, so the total number of

taken edges is at most 8n(b+ 1).

The second stage. Recall that at the second stage Maker has to connect

in both directions every two disjoint sets A,B of size (1−o(1))n

(lnn)2/5.

Consider a random tournament obtained from Kn by directing each edge

uniformly and independently of the other edges. For every two disjoint sets

of vertices A,B, the number of edges from A to B is binomially distributed.

Denote by e(A,B) the number of edges from A to B and by e(B,A) the

number of edges from B to A. By the Chernoff bound (see, e.g., [13]), we

have

Pr

[∣∣∣∣e(A,B)− |A||B|2

∣∣∣∣ ≥ ε|A||B|]≤ e−ε

2|A||B|/2,

and similar inequality holds also for e(B,A).

Therefore, if |A| = |B| = (1−o(1))n

(lnn)2/5the probability that e(A,B) or e(B,A)

is greater than 12·|A||B|(1+n−1/2+o(1)) is 2−2n, and hence by the union bound

for every two such sets, there are at least 12· |A||B|(1 +n−1/2+o(1)) ≥ 0.99n2

2(lnn)4/5

in each direction. Fix a tournament T ∗ with this property.

At the second stage, Maker always directs edges that agree with T ∗. That

is, he can only direct an edge from u to v if (u, v) ∈ E(T ∗). For every two

such sets, at most 8n(b + 1) ≤ 12n2

lnnedges were directed at the first stage of

the game, and hence at the beginning of the second stage at least 0.99n2

2(lnn)4/5

edges that are directed from A to B in T ∗ are unclaimed.

Now Maker and Breaker switch roles. Maker clearly wins if he prevents

Breaker from claiming all the edges from a set A to a set B, where |A| =

|B| = k = n(lnn)2/5

. To this end, we apply the Beck-Erdos-Selfridge criteria

(Theorem 6.3), with p = b = n(1+o(1))lnn

, q = 1, the size of each hyperedge is at

least 0.99n2

2(lnn)4/5and the total number of sets is at most

(nk

)2. We have

6.5 The H-creation game 109

∑A∈F

(q + 1)−|A|p <

(n

k

)2

· (q + 1)−|A|p

<(enk

)2k

· 2−n lnn

3(lnn)4/5

≤ 24n·log logn

(lnn)2/5 · 2−n(logn)1/5

6 1.

Therefore in our case Maker wins and hence every two sets of size k are

connected in both ways.

We conclude that at the end of the second stage Maker has a strongly

connected graph, and hence by the end of the game the obtained tournament

is strongly connected, and Maker wins. This proves Item 2 in Theorem 6.2.

6.5 The H-creation game

In this game, a fixed oriented graph H is given. Maker wins if the obtained

tournament contains a copy of H, and Breaker wins otherwise. Note that if

H does not contain a directed cycle then Maker surely wins for large enough

n, as every tournament of size n contains a transitive tournament of size

log n, which contains H as a subgraph.

Our starting point is an upper bound on the bias threshold for a general

fixed graph H. Given a directed graph H, and a permutation σ : V (H) →|V (H)|, we define the feedback arc set of H with respect to σ as

FAS(H, σ) = |(u, v) ∈ E(H) : σ(u) > σ(v)|.

In words, this parameter measures the number of edges that are going in the

wrong direction with respect to σ. Let

FAS(H) = minσFAS(H, σ).

This is the minimal number of edges of H that has to be deleted in order to

make H an acyclic graph. If, for example, H is a random tournament on t


vertices, then it is easy to show that with high probability FAS(H) is close

to t(t− 1)/4.

We have the following upper bound.

Lemma 6.11. Let H be a graph on t vertices, and let r = FAS(H). Suppose

that Maker and Breaker play an orientation game on Kn. Then if b > c(H) ·nt/r then Breaker has a strategy guaranteeing that the obtained tournament

does not contain a copy of H, where c(H) > 0 depends only on H.

Proof. The proof follows by a simple application of the Beck-Erdos-Selfridge

theorem (Theorem 6.3). Breaker will choose an arbitrary permutation σ of

the vertices, and at every turn he will direct the edges according to σ. That

is, whenever he chooses to direct an edge uv, and σ(v) > σ(u) the edge will

be directed from u to v. Hence, if Maker creates a copy of H, by definition

he orients at least FAS(H) edges in the opposite direction with respect to σ.

We can thus reduce the game to the classical Maker-Breaker game as follows.

In every set of t vertices, Maker can win only if he claims at least r edges

that are induced by this set, and Breaker wins if he prevents Maker from

doing so. The total number of winning sets for Maker is at most(nt

)·((t2)r

).

Therefore, if (q + 1)r = Ω((nt

)·((t2)r

)) then by Theorem 6.3, Breaker has a

winning strategy. This is the case if b > c(t, r) · nt/r, and hence the lemma

holds.

It is worth noting that following the methods of Bednarska and Luczak [24],

one can prove that if b = O(n|V (H)|−2|E(H)|−1 ) then Maker has a winning strategy as

follows. Maker chooses at each round a random undirected edge and orients

it randomly, independently of the other choices. Roughly speaking, one can

show that by the end of the game the obtained graph looks random in some

sense, and hence if the bias is large enough then with high probability it

contains a copy of H.

However, following their approach does not give sharp bounds in our case.

To see this, observe that their results give a sharp bound of b = Θ(√n)) for

the triangle creation game in the classical Maker-Breaker settings, while in

orientation games the correct bias for creating a cyclic triangle or even any

directed cycle is b = Θ(n), as we will see shortly.


We next generalize the result of Section 6.3 and show that in the case

that H is a fixed cycle, Maker wins even if b = Ω(n).

Proposition 6.12. For every constant k ≥ 3 there is a constant γ(k) > 0

such that if b < γ(k) · n then Maker wins the b-biased Ck-creation game.

Proof. We first observe that if a tournament T contains a cycle of length

k+(k−2)r for some r ∈ N then it also contains a cycle of length k. The proof

of this observation is by induction. It is trivially true for r = 0. Suppose this

is true for all values smaller than some fixed r, and let v1, v2, . . . , vk+(k−2)r, v1

a cycle of length k + (k − 2)r. Consider the edge between vk and v1. If the

edge is directed from vk to v1 there is a cycle of length k and we are done.

Otherwise, vk, vk+1, . . . , vk+(k−2)r, v1, vk is a cycle of length k+ (k− 2)(r− 1)

and therefore by the induction hypothesis T contains a cycle of length k, as

required.

Therefore, in order to create a cycle Ck, Maker has to create some cycle

of length k + (k − 2)r. By Lemma 6.4, at each round Maker can extend

a longest directed path by 1. Maker’s strategy is to close a cycle of length

k + (k − 2)r whenever it is possible, and to extend a longest directed path

by 1 if it is not possible. After t rounds, there is a path of length t, and we

denote it by x1, . . . , xt. For every i ≥ k, the number of edges from xi to xj,

j < i, that may close a cycle of length k+ (k− 2)r is at least ik−2− 2. Hence

the total number of edges that may close a cycle of length k + (k − 2)r for

some r is at least

t∑i=k

(i

k − 2− 2

)≥ (t+ k)(t− k)

2(k − 2)− 2t.

Therefore, the number of such edges is at least(t2)k

for t = Ω(k2), that is

at least (1/k)-fraction of the edges for such t. Among these(t2)k

edges, at

most (b + 1)t were oriented by either Maker or Breaker in previous rounds.

Note that as long as the bias b is smaller than n/2, the game lasts at least

t = n rounds, and results in a path of length n−1, unless Maker wins before.

Therefore if b < n−1−2k2k

then Maker wins the game, as required.


Recall that a tournament T is k-colorable if its edges can be partitioned

into k transitive tournaments. Berger et al. [35] studied the class of tour-

naments H with the property that there is constant c(H) such that every

H-free tournament T is c(H)-colorable. They called every such tournament

H a hero, and characterized the set of such tournaments. We next show that

for every k > 0 Maker has a strategy to create a non k-colorable tournament

as long as the bias is a sufficiently small linear function of n.

Lemma 6.13. Let k > 0, and suppose that b = cnk log k

for some sufficiently

small constant c > 0. Then Maker has a strategy to create a non k-colorable

tournament.

Proof. It is rather easy to see using a Chernoff bound (as it was done in

Section 6.4) that a random tournament obtained by directing each edge uni-

formly and independently of the other choices has typically the following

property. For every ordered pair of disjoint sets A,B of size n/2k, there are

Θ(n2

k2) edges in each direction between A and B. Fix a tournament T ∗ with

this property.

Define a hypergraph H whose vertices are the edges of T ∗ and whose

edges are all the edges from A to B in T ∗ for every ordered pair A,B of size

n/2k. Maker will win the game by orienting one edge from every hyperedge

in H according to T ∗. To this end, Maker will play to prevent Breaker from

orienting all the edges in some hyperedge from H. By the end of the game,

there is an edge between every two sets of size n/2k and hence the obtained

tournament does not contain an acyclic set of size n/k, and therefore is not

k-colorable.

There are(

nn/2k

)2 ≤ (2ek)n/k choices of ordered pairs (A,B), each cor-

responds to a hyperedge of H. The size of each hyperedge is Θ(n2

k2). By

applying the Beck-Erdos-Selfridge strategy (Theorem 6.3, with Maker play-

ing role of Breaker, p = b and q = 1), if b = cnk log k

then Maker has a winning

strategy, as required.

A simple consequence of Lemma 6.13 is the following generalization of

Lemma 6.4. Berger et al. [35] provided a list of five minimal tournaments

H1, H2, . . . H5, and proved (Theorem 5.1 in [35]) that every non-hero tourna-


ment must contain at least one of H1, . . . , H5 as a subtournament. For every

1 ≤ i ≤ 5, one can check that FAS(Hi) ≥ 2.

Consider any oriented graph H with FAS(H) = 1. Let σ be an ordering of

V (H) with a single edge that does not agree with σ. Let H ′ be a tournament

on V (H) that contains H as subgraph and is defined as follows. For every two

vertices u, v ∈ V (H), if (u, v) ∈ E(H) we let (u, v) ∈ E(H ′). If (u, v), (v, u) /∈E(H), we let (u, v) ∈ E(H ′) if σ(v) > σ(u) and (v, u) ∈ E(H ′) otherwise.

We get that FAS(H ′) = 1 as well.

Clearly, it is sufficient to construct a copy of H ′ for Maker’s win. The

result of Berger et al. [35] can be applied only for tournaments, and hence

we will use it to show that Maker can construct a copy of H ′.

Since FAS(H ′) = 1 then H ′ is a hero, and therefore every tournament that

does not contain a copy of H ′ is c(H ′)-colorable, where c(H ′) is a constant

that depends only on H ′. By Lemma 6.13, if b = Θ(n) then Maker has a

strategy so the obtained tournament is not c(H ′)-colorable. We therefore

have the following.

Proposition 6.14. For every oriented graph H with FAS(H) = 1 there is a

constant γ(H) > 0 such that Maker wins the γn-biased H creation game.

We conjecture that the bias threshold that guarantees Maker’s win strongly

depends on FAS(H). It will be interesting to find further quantitative results

in this direction.


Chapter 7

Local Rainbow Colorings


Given a graph H, we denote by C(n,H) the minimum number k such

that the following holds. There are n colorings of E(Kn) with k-colors, each

associated with one of the vertices of Kn, such that for every copy T of H

in Kn, at least one of the colorings that are associated with V (T ) assigns

distinct colors to all the edges of E(T ).

We characterize the set of all graphs H for which C(n,H) is bounded by

some absolute constant c(H), prove a general upper bound and obtain lower

and upper bounds for several graphs of special interest. A special case of our

results partially answers an extremal question of Karchmer and Wigderson

motivated by the investigation of the computational power of span programs.

7.1 Introduction

Consider the following question, motivated by an extremal problem suggested

by Karchmer and Wigderson, see [112]. Given a fixed graph H, let C(n,H)

denote the minimum number k such that there is a set of n colorings fv :

E(Kn) → [k] : v ∈ V (Kn), with the following property. For every copy T

of H in Kn, there is a vertex u ∈ V (T ) so that fu is a rainbow coloring of

E(T ), that is, no two edges of T get the same color by fu. A set of colorings

that satisfies this condition is called an (n,H)-local coloring. Determine or

116 Local Rainbow Colorings

estimate the function C(n,H).

For example, if T is a path with 2 edges, two colors are sufficient for

every n. Indeed, simply define fv(uw) = 1 if v ∈ u,w, and fv(uw) = 2

otherwise. Obviously, this is a legal local coloring, and therefore we conclude

that C(n, P2) = 2 for all n ≥ 3. As we show in this chapter, for almost every

graph T , every (n, T )-local coloring requires a number of colors that grows as

n grows. In fact, even the path P3 of 3 edges requires more than a constant

number of colors.

Motivation. Karchmer and Wigderson [77] defined the Span Program com-

putational model. This is a linear algebra computational model. The pro-

gram is defined by a set of vectors A over GF2, where each vector is marked

with a literal. Every input of the program corresponds to the subset of vec-

tors of A whose marking literals have value 1 in the given input, and the input

is accepted if and only if this subset spans some fixed vector, say the all-ones

vector. It turns out that this model is stronger than other known models

like Switching Networks and DeMorgan Formulas (see [75]), and therefore

lower bounds for the minimum possible size of Span Programs that compute

explicit Boolean functions are desirable.

The Fusion Method of Karchmer and Wigderson [77], [112] enables one to

prove lower bounds for this model using certain extremal results. With this

motivation in mind, Wigderson suggested in [112] three problems. One of

them is the following (also suggested as a research problem in [75], Chapter

16, Problem 9).

Problem 7.1. Let k be the minimum number for which the following holds.

There exist n colorings c1, c2, . . . , cn of the n-cube 0, 1n in k colors such

that for every triple of distinct vectors x, y, z there is a coordinate i on which

not all three vectors agree and the three colors ci(x), ci(y), ci(z) are pairwise

distinct. Determine or estimate the smallest number k for which such a

collection of colorings exist.

In [77] it is shown that k does have to grow with n, proving that it is at

least Ω( log log∗ nlog log log∗ n

), where log∗ n is the minimum number m so that starting

with n one gets a number that does not exceed 1 by iteratively applying

7.1 Introduction 117

the function log2(x) m times. As a special case of our results we obtain

a far better lower bound, proving that for the path with 3 edges P3, every

(n, P3)-local coloring requires Ω(( lognlog logn

)1/4) colors. It is easy to see that this

lower bound holds for the number of required colors in the problem above as

well, even if one considers only vectors of Hamming weight 2. Unfortunately,

it does not seem that this improved bound yields any new results for span

programs.

Our Results. Our first result characterizes the set of all graphsH for which

C(n,H) is bounded by some absolute constant that depends only on H. To

this end, we define 2-locally large graphs, and show that C(n,H) ≤ c(H)

for every n if and only if H is not 2-locally large. This implies the following

somewhat surprising corollary.

Theorem 7.1. For a fixed graph H there is a constant c(H) so that C(n,H) ≤c(H) for every n if and only if H contains at most 3 edges and H is neither

P3 nor P3 together with any number of isolated vertices. Moreover, in all

these cases C(n,H) ≤ 5 for every n.

We next consider upper bounds. We start with a simple general upper

bound for every fixed graph H, obtained by applying the local lemma. this

shows that for every fixed graph H, C(n,H) is sublinear in n.

Theorem 7.2. Let H be a fixed graph with r vertices. Then C(n,H) ≤O(n

r−2r · r4).

Note that the most difficult case here is when H is a complete graph on r

vertices, though this general approach improves the bound for sparser graphs

only by a constant that depends on r.

We also provide a simple explicit construction for the case H = P3. This

construction is summarized in the following proposition.

Proposition 7.3. There is an explicit construction showing that C(n, P3) ≤2d√n e.

In the last part of this chapter we obtain lower bounds for several graphs,

including a (log n)Ω(1) lower bound for C(n,H) for H = P3, for any star


with t ≥ 4 leaves, and for any graph consisting of t ≥ 4 independent edges.

This implies a similar lower bound for any fixed graph with sufficiently many

edges.

Theorem 7.4. Let Pt denote the path with t edges, It the graph consisting of

t independent edges and St the star with t edges. The following lower bounds

hold.

1. C(n, P3) = Ω(( lognlog logn

)1/4).

2. C(n, I4) = Ω(n1/6) and C(n, It) = Ω(n1/4) for t ≥ 5.

3. C(n, S4) = Ω(n1/4) and C(n, St) = Ω(n1/3) for t ≥ 5.

4. C(n, P7), C(n, P8) = Ω(n1/6) and C(n, Pt) = Ω(n1/4) for any fixed t ≥9.

Observe that if H ′ ⊂ H is a subgraph of H on the same set of vertices then

every lower bound for C(n,H ′) implies the same lower bound for C(n,H).

As every large enough graph H (in terms of the number of edges) contains a

star with 4 edges or 4 independent edges, and as it is possible to deal with

isolated vertices in these cases, one can prove an nΩ(1) lower bound for any

fixed graph with sufficiently many edges. This is stated in the following.

Theorem 7.5. For any graph H with at least 13 edges there is a constant

b = b(H) > 0 so that C(n,H) ≥ Ω(nb).

7.2 Preliminaries

Let Pt denote the path with t edges, St the star with t leaves (and hence, t

edges), and It the graph consisting of t independent edges.

Let Kn denote the complete graph on n vertices, and K(t)n the hypergraph

containing all subsets of size t of [n]. In the course of the proofs we will apply

the following hypergraph Ramsey result for 3-uniform hypergraphs, due to

Erdos and Rado [60].

7.3 Graphs requiring a bounded number of colors 119

Theorem 7.6. There is an absolute constant c such that if n > 22ck log(k)rthe

following holds. For every k coloring of E(K(3)n ) there is a set T ⊆ V (K

(3)n )

of size r, such that A ∈ E(K(3)n ) : A ⊂ T is monochromatic.

Given a graph G, we denote by ∆(G) the maximum degree of a vertex

in G. A proper edge coloring of a graph G is a mapping ϕ : E(G) → [k]

such that every two adjacent edges get distinct colors. The edge chromatic

number (or chromatic index) of any simple graph G is the minimum k for

which such a coloring exists. Vizing’s Theorem (see, e.g., [45]) states that the

edge chromatic number of any simple graph G is either ∆(G) or ∆(G) + 1.

7.3 Graphs requiring a bounded number of

colors

Our goal in this section is to prove Theorem 7.1. We start with some auxiliary

definitions.

Let H be a fixed graph on r vertices, and let σ : V (H) → [r] be a

permutation. Roughly speaking, H is 2-locally large if for every vertex there

are two edges that ”look the same” with respect to σ. More formally, for

every vertex v ∈ V (H), define the following five types of edges.

1. T 1v = (u, v) : σ(u) > σ(v).

2. T 2v = (u, v) : σ(u) < σ(v).

3. T 3v = (u1, u2) : σ(v) < σ(u1) < σ(u2).

4. T 4v = (u1, u2) : σ(u1) < σ(v) < σ(u2).

5. T 5v = (u1, u2) : σ(u1) < σ(u2) < σ(v).

Thus there are two types of edges that are incident to v, backward edges

and forward edges (with respect to σ). In addition, there are three types of

non-incident edges, again with respect to σ. There are edges before v, edges

after v, and edges that cross v.


Definition. A graph H on r vertices is 2-locally large if there is a permu-

tation σ : V (H)→ [r] such that for every v ∈ V (H), at least one of the sets

T 1v , . . . , T

5v has at least two elements.

We can now state the main result of this section.

Theorem 7.7. Let H be a fixed graph. Then C(n,H) ≤ c(H) for every n if

and only if H is not 2-locally large. In this case C(n,H) ≤ 5 for all n.

Before presenting the proof of this theorem, we turn to list the graphs

which are not 2-locally large. It is obvious that every graph with at least 6

edges is 2-locally large. A path of length 3, denoted by xyzw is also 2-locally

large by letting σ(x) < σ(z) < σ(y) < σ(w). It is also easy to prove by

inspection that every graph with at most 3 edges besides the path P3 is not

2-locally large. We next claim that actually every graph with at least 4 edges

is 2-locally large.

Proposition 7.8. Let H be a graph with at least 4 edges. Then H is 2-locally

large.

Proof. It is sufficient to prove the assertion for graphs with exactly 4 edges

and no isolated vertices, as the property of being 2-locally large is maintained

under addition of edges or isolated vertices. Consider the following cases.

• H has 4 independent edges, denoted by (u1, u2), (u3, u4), (u5, u6), (u7, u8).

In this case let σ(ui) < σ(uj) for every i < j and we are done.

• H is the union of two paths with two edges each. Denote these paths by

u1u2u3 and v1v2v3. Then by taking σ(ui) < σ(vj) for every 1 ≤ i, j ≤ 3,

we get that every vertex has two indistinguishable edges.

• H is the union of a path with two edges, denoted by xyz with 2 in-

dependent edges, denoted by (u1, u2) and (u3, u4). In this case let

σ(x) < σ(y) < σ(z) < σ(u1) < σ(u2) < σ(u3) < σ(u4) and we are

done.

• H contains a connected component with 3 edges and a single indepen-

dent edge (u1, u2). In this case the connected component is either a

path of length 3 (which is 2-locally large), a triangle or a star with 3

7.3 Graphs requiring a bounded number of colors 121

leaves. In the last two cases we let σ(u1) = 1, σ(u2) = |V (H)| and by

symmetry it is easy to see that for every mapping of the vertices of the

connected component and for every vertex v we have |(T iv)∩E(H)| > 1

for some i.

• H is connected (and has 4 edges). In this case H contains a path of

length 3 or a star with 4 leaves, and in the last case every embedding

will suffice.

Proof of Theorem 7.1. By Theorem 7.7 C(n,H) grows as n grows if and

only if H is 2-locally large, and otherwise it is bounded by 5. By Propo-

sition 7.8 these graphs are exactly all graphs with at least 4 edges and P3.

Theorem 7.1 follows.

Proof of Theorem 7.7. We start by proving that for every graph H that is

not 2-locally large, C(n,H) is bounded by 5. Indeed, let Kn be the complete

graph on n vertices, and fix an arbitrary mapping σ : V (Kn) → [n]. For

every vertex v ∈ Kn, we let fv(xy) = i if and only if xy ∈ T iv.Let T be a subgraph of Kn isomorphic to H. Since T is not 2-locally

large, there is a vertex v ∈ V (T ) such that |(T iv)∩E(H)| ≤ 1 for every i, and

hence fv(e1) 6= fv(e2) for every e1, e2 ∈ E(T ), as required.

Suppose now that H is 2-locally large. Let σ : V (Kn) → [n] be an

arbitrary bijection, and let f1, . . . , fn : E(G) → [k] be n colorings, one for

each vertex. We define a coloring ofK(3)n , the complete 3-uniform hypergraph,

which is defined with a slight abuse of notation on the same set of vertices as

Kn. For every three vertices u1, u2, u3 ∈ V (Kn), with σ(u1) < σ(u2) < σ(u3)

we define the coloring of the hyperedge (u1, u2, u3) as the following ordered

nine-tuple.

[fu1(u1u2), fu1(u1u3), fu1(u2u3), fu2(u1u2), fu2(u1u3),

fu2(u2u3), fu3(u1u2), fu3(u1u3), fu3(u2u3)].

q


This is a coloring of E(K(3)n ) with k9 colors. By Theorem 7.6, if n >

229ck9 log kr

, there is a monochromatic set of vertices Q of size r (namely, all

3-edges that are contained in Q have the same color). Consider a copy T of

H in the vertices of Q such that for every vertex v in T , there is 1 ≤ i ≤ 5

such that |(T iv) ∩ E(H)| > 1 (observe that there is such a copy since T is

2-locally large).

We complete the proof by showing that T does not admit a rainbow

coloring fv for any v ∈ V (T ). Take a vertex v ∈ V (T ). Let e1, e2 ∈ E(T )

such that e1, e2 ∈ T iv for some 1 ≤ i ≤ 5. Since the two hyperedges that

contain v, e1 and v, e2 are colored the same, we conclude that fv(e1) = fv(e2).

Therefore, every vertex v ∈ T has two edges which get the same color, and

we conclude that Kn is not (H, k)-locally colorable, as required.

Note that the above proof shows that for every fixed H which is 2-locally

large, C(n,H) ≥ cH( log lognlog log logn

)1/9.

7.4 Upper bounds

In this section we prove a simple upper bound on C(n,H) for every H, and

also provide an explicit construction for the specific case of H = P3.

Proof of Theorem 7.2. Clearly it is enough to prove the theorem for

the case that H is a complete graph on r vertices, as if fv assigns pairwise

distinct colors to the set of edges of Kr, it also gives distinct colors for every

subgraph of Kr.

Let k = nr−2r · r4. For every v ∈ V (G), e ∈ E(G), let fv(e) be uniformly

distributed among the k possible colors, independently of the other choices.

We will prove that this random coloring is good with a positive probability

by applying the symmetric case of the Local Lemma (see, e.g., [13], Chapter

5).

Indeed, given a fixed graph T that is isomorphic to H, and a vertex

v ∈ V (T ), the probability that two edges in T get the same color in fv is

clearly bounded by (r2

)· ((r2

)− 1)

2k≤ r4

8k.

7.4 Upper bounds 123

Denote by A(T ) the event that all the colorings fv, for every v ∈ V (T ), do

not assign pairwise distinct colors to all edges in E(T ). Since the colorings

are chosen independently, we conclude that Pr [A(T )] ≤ ( r4

8k)r.

Observe that each event A(T1) is mutually independent of all other events

A(T2) besides those for which T1, T2 share at least one edge, and therefore at

least two vertices. Thus, for every T , A(T ) is independent of all but at most(n−2r−2

)< nr−2 other events. We have

nr−2 · ( r4

8k)r · e < 1.

Hence, by the Local Lemma (see, e.g., [13], Chapter 5), we get that with

positive probability non of the events A(T ) occurs, and hence a good coloring

exists. We next show an explicit construction for the case of H = P3. This

explicit construction is slightly better than the randomized one (by a constant

factor).

Proof of Proposition 7.3. We will find a set of n colorings of the edges of

Kn by 2d√n e colors, such that in each copy of P3 at least two of the vertices

assign distinct colors to all edges. By Vizing’s theorem, Kn admits an n-edge

coloring ϕ, where no two adjacent edges get the same color. Denote these

colors by 1, 2, . . . , n, and let s = d√n e. Clearly, for every 1 ≤ q′ < q′′ ≤ n,

we have dq′/se 6= dq′′/se or q′ mod s 6= q′′ mod s.

We next define fv(ab) for every v ∈ V (Kn) by a pair < x, y > with x ∈0, 1 and y ∈ [s] = 1, 2, . . . , s as follows. The first coordinate determines

whether ab is incident with v or not (that is, x = 1 if either a = v or b = v,

and x = 0 otherwise). Let r = ϕ(ab). For the second coordinate, if x = 1 we

define the second coordinate of fv(ab) as y = dr/se, and if x = 0 we define

it as y = r mod s.

Let abcd be a copy of P3 in Kn. Then fb(ab), fb(bc) 6= fb(cd) and

fd(ab), fd(bc) 6= fd(cd), as their first coordinates differ. Now, let r1 = ϕ(ab)

and r2 = ϕ(bc). Since ϕ is a legal edge coloring, r1 6= r2 and therefore

dr1/se 6= dr2/se or r1 mod s 6= r2 mod s. Therefore fb(ab) 6= fb(bc) or

fd(ab) 6= fd(bc) and we get at least one good coloring of abcd. The same


argument shows that also either fa or fc defines a good coloring of abcd, and

the proposition follows.

7.5 Lower bounds

In this section we obtain several lower bounds which are significantly better

than the general lower bound that follows from the proofs of Theorem 7.1

and Theorem 7.7.

We start with a (log n)Ω(1) lower bound for H = P3.

Proof of Item 1 in Theorem 7.4. Let Kn be the complete graph on

n > k(k2+1)2 + k2 + 2 vertices, and let Y be an arbitrary set of vertices of size

k2 + 1 and X = V (Kn) \ Y . Fix a coloring fv of the edges of Kn for every

v ∈ V (Kn). For every vertex x ∈ X, define a vector with |Y |2 coordinates,

by letting x[u1, u2] = fu1(xu2) for every u1, u2 ∈ Y . By the pigeonhole

principle, there are two vertices x1, x2 ∈ X such that the vector corresponding

to x1 equals that corresponding to x2. By applying again the pigeonhole

principle, there are two vertices y1, y2 ∈ Y such that fx1(x2y1) = fx1(x2y2)

and fx2(x2y1) = fx2(x2y2). It follows that the path x1y1x2y2 does not have a

rainbow coloring. Therefore, if there are good colorings f1, f2, . . . , fn for Kn

then n < k(k2+1)2 + k2 + 2 and hence k = Ω(( lognlog logn

)1/4), as required. We proceed with an auxiliary lemma that will be used in proving lower

bounds for the other cases.

Lemma 7.9. Let fv : v ∈ V (K(t)2n ) be a set of 2n colorings of V (K

(t)2n ) with

k colors. Suppose that for every e ∈ E(K(t)2n ) there is a vertex v ∈ e such that

fv assigns distinct colors to all the vertices of e.

1. If t = 4 then k ≥ (1− o(1))n1/3.

2. If t ≥ 5 then k ≥ (12− o(1))n1/2.

Proof. Let A,B ⊆ V (K(t)2n ) be a partition of the vertices into two disjoint

sets of size n. For every a ∈ A, the vertices of B are partitioned into at most

k sets according to their color in fa. We proceed by counting the number of

7.5 Lower bounds 125

pairs of elements in B that are colored the same by some vertex in A (with

multiplicities). By convexity, every vertex a ∈ A contributes

k∑i=1

(|f−1a (i)|

2

)≥ k ·

(n/k

2

)=n2

2k· (1− k/n).

Therefore, there is a pair of vertices b′, b′′ ∈ B that are colored the same by

at leastn3

2k(|B|

2

) · (1− k/n) ≥ n

k− 1

members of A.

Suppose that t = 4. If k < n1/3(1− o(1)) we conclude that there is a set

A′ ⊆ A of size k2 + 1 and two elements b′, b′′ ∈ B such that for every a ∈ A′we have fa(b

′) = fa(b′′). By the pigeonhole principle, among the elements of

A′ there is a set A′′ of size k + 1 whose members are mapped to the same

value by fb′ . Similarity, there are two elements from A′′, denoted by a∗ and

a∗∗ that are mapped to the same value by fb′′ , and we conclude that the set

b′, b′′, a∗, a∗∗ does not admit a good coloring, and therefore k ≥ n1/3(1−o(1))

as required.

Suppose, now, that t ≥ 5 and k < 12· n1/2(1− o(1)). Then there is a set

A′ of size 4k+ t+1 and two elements b′, b′′ ∈ B such that for every a ∈ A′ we

have fa(b′) = fa(b

′′). Applying iteratively the pigeonhole principle, we get

that there is a set X ′ of 2k+t3

triplets of elements in A′, where the members

of each triple have the same fb′ image, and there is a set X ′′ of 2k+t3

triplets

of elements in A′ so that the members of each triple have the same fb′′

image. Therefore there are two triplets, one from X ′ and one from X ′′ that

intersect, denote them by (x′1, x′2, x′3) (from X ′) and (x′1, x

′′2, x

′′3) (from X ′′).

Without loss of generality assume that x′2 6= x′′2. If t = 5, we get that the

set b′, b′′, x′1, x′2, x′′2 does not admit a good coloring and we are done. If t > 5,

add to this set t − 5 arbitrary elements from X ′, and again we get a set of

size t that does not admit a good coloring. Therefore k ≥ 12· n1/2(1 − o(1))

and the lemma follows.

We proceed with the proofs of the remaining items of Theorem 7.4.


Proof of Item 2 in Theorem 7.4. LetKn be the complete graph on n ver-

tices, and suppose for simplicity that n is even. Let (v1, v2), (v3, v4), . . . , (vn−1, vn)

be n/2 independent edges. Fix a set of colorings fv1 , . . . , fvn with k colors

such that for every copy of It there is a vertex in that copy that assigns

distinct colors to all edges of this copy. Consider the complete hypergraph

K(t)n/2, where each vertex is associated with one of the n/2 independent edges.

Every vertex x ∈ V (K(t)n/2) that corresponds to an edge (v2i−1, v2i) defines a

coloring of V (K(t)n/2) with k2 colors, as the combination of fv2i−1

and fv2i . By

the definition of local coloring, for every e ∈ E(K(t)n/2) there is v ∈ e that gives

all the vertices in e distinct colors. Thus by Lemma 7.9 k2 ≥ n1/3(1− o(1))

(if t = 4) and k2 ≥ n1/2(12− o(1)) (if t ≥ 5), and the result follows.

Proof of Item 3 in Theorem 7.4. Fix a set of n colorings of Kn such

that for every star St, there is a vertex that assigns distinct colors to all

edges of the star. Fix an arbitrary vertex x ∈ V (Kn), then by the pigeonhole

principle there is a set S ⊆ V (Kn) of size (n− 1)/k such that fx(xu) is the

same for every u ∈ S. Consider the complete hypergraph K(t)(n−1)/k, where

each vertex is associated with one of the vertices of S, and define a set of

(n− 1)/k k-colorings gv : v ∈ S of the hypergraph by gv(u) = fv(xu). By

definition, for every e ∈ K(t)(n−1)/k there is v ∈ e that gives distinct colors for

all the vertices of e, and thus by Lemma 7.9 we have k ≥ (1− o(1))(nk)1/3 (if

t = 4) and k ≥ (1/2 − o(1))(nk)1/2 (if t ≥ 5). Hence we get k = Ω(n1/4) in

the first case and k = Ω(n1/3) in the second case, as required.

Finally, we get a bound for the case of a path of length at least 7 as an

immediate corollary.

Proof of Item 4 in Theorem 7.4. Every path of length t contains a

spanning subgraph consisting of dt/2e independent edges. Therefore, the

desired result follows from Item 2.

Proof of Theorem 7.5. By Vizing’s Theorem, any graph H with at least

13 edges contains either a vertex of degree at least 4 or 4 independent edges.

Therefore, it contains a spanning subgraph consisting of a vertex of degree 4

7.6 Concluding remarks and open questions 127

and a set of some r isolated vertices, or a spanning subgraph consisting of 4

independent edges and a set of q isolated vertices. In the first case we fix r

vertices v1, . . . , vr and another vertex x, and apply the pigeonhole principle

to find a set of (n − r − 1)/kr vertices uj 6= x, v1, . . . vr, so that for each

vertex vi, fvi(xuj) is constant for all uj. We then apply the argument in the

proof of Item 3 to the star consisting of all edges xuj. The second case is

similar. We first fix q vertices vi and find a set of n−q2kq

independent edges

on the other vertices that are colored the same by each vertex vi. We then

apply the argument in the proof of Item 2 to this set.

7.6 Concluding remarks and open questions

We introduced the notion of (n,H)-local colorings and studied various general

and concrete bounds on C(n,H)- the minimum number of colors that are

required such that there are n colorings of Kn so that for every copy of H,

one of the vertices assigns distinct colors to all edges of that copy. This

generalizes and partially answers a question of Karchmer and Wigderson.

We characterized the (small) set of graphs that require a constant number

of colors. It would be interesting to decide whether every graph apart from

the members of this set requires a polynomial number of colors. We can

strengthen the assertion of Theorem 7.5 and show that it holds with the

same b for every H. More precisely, there are absolute positive constants b

and c such that for any graph H with at least c edges there is a constant

a(H) > 0 so that C(n,H) ≥ a(H)nb for every n. We omit the detailed

argument.

The problem of improving the lower bound for the case of H = P3 is

interesting as well, and will improve the lower bound for the Karchmer-

Wigderson question. Also, it would be nice to obtain better upper bounds

by exhibiting (probabilistic or explicit) colorings.

Finally, there are gaps between the upper and lower bounds of C(n,H)

for almost every H, and closing the gap in each of the considered cases might

need additional ideas. In particular, it would be nice to decide if for every

ε > 0, there is an r = r(ε) so that C(n,Kr) ≥ n1−ε for all sufficiently large

n.


Chapter 8

Perfectly balanced partitions of

smoothed graphs


For a graph G = (V,E) of even order, a partition (V1, V2) of the vertices

is said to be perfectly balanced if |V1| = |V2| and the numbers of edges in

the subgraphs induced by V1 and V2 are equal. For a base graph H define

a random graph G(H, p) by turning every non-edge of H into an edge and

every edge of H into a non-edge independently with probability p. We show

that for any constant ε there is a constant α, such that for any even n

and a graph H on n vertices that satisfies ∆(H) − δ(H) ≤ αn, a graph G

distributed according to G(H, p), with εn≤ p ≤ 1 − ε

n, admits a perfectly

balanced partition with probability exponentially close to 1. As a direct

consequence we get that for every p, a random graph from G(n, p) admits a

perfectly balanced partition with probability tending to 1.

8.1 Introduction

Given a graph G of even order, an equipartition (V1, V2) is perfectly balanced

if the number of edges spanned by G[V1] equals the number of edges spanned

by G[V2]. This definition can be extended to graphs of odd order, and in this

case we require that the number of vertices and edges in each part differ by

130 Perfectly balanced partitions of smoothed graphs

at most 1.

It is not difficult to verify that there are graphs that do not admit perfectly

balanced partitions. The simplest example is the star K1,n−1. Caro and

Yuster [50] proved that for every graph G = (V,E) with n2−ε edges for some

ε > 0 there are two disjoint sets S1, S2 ⊆ V , |S1| = |S2| = n/2 − o(n) such

that the subgraphs induced by S1 and S2 have exactly the same number of

edges, and this is asymptotically tight.

It is natural to seek for families of graphs that admit perfectly balanced

partitions. In this chapter we show that smoothed graphs admit such a

partition almost surely. Namely, for a given εn≤ p ≤ 1 − ε

n, let H be a

graph on n vertices that satisfies ∆(H)− δ(H) ≤ αn for some small constant

α(ε) (here ∆(H) is the maximal degree and δ(H) is the minimal degree of

H). If we modify every edge and non-edge independently with probability p,

almost surely we get a graph that admits a perfectly balanced partition. As

a corollary, we get also that for every p = p(n), a graph distributed according

to G(n, p) admits a perfectly balanced partition almost surely.

Smoothed Graphs. Smoothed analysis studies the behavior of objects

after adding a small amount of randomness. This concept was introduced

in the context of algorithms by Spielman and Teng [105], and was studied

also in the context of graphs (see, for example, [40, 41, 85]) and hyper-

graphs [107]. Given a fixed graph H on n vertices, define the probability

distribution G(H, p) as follows. Every edge of H is deleted with probability

p, and every pair of non-adjacent vertices of H is connected by an edge with

probability p, all these events are mutually independent. We will usually be

interested in the case that p is relatively small, thus one may consider graphs

from this distribution as noisy (or smoothed) variations of H.

We can now state the main result of this chapter.

Theorem 8.1. For every ε > 0 there are α(ε) > 0 and c(ε) > 0 such that

the following holds for every even n. For every graph H on n vertices that

satisfies ∆(H)− δ(H) ≤ αn, a graph G distributed according to G(H, p) for

some εn≤ p ≤ 1− ε

nadmits a perfectly balanced partition with probability at

least 1− e−cn.

8.2 Proofs 131

Here we state the result and provide a proof for graphs of even size. It

is not difficult to extend it to graphs of odd size. We note that the theorem

is essentially tight: A graph distributed according to G(K1,n−1,1

2n) admits a

perfectly balanced partition with probability o(1).

Theorem 8.1 implies the following corollary.

Corollary 8.2. Let n be an even number. A graph distributed according to

G(n, p) admits a perfectly balanced partition asymptotically almost surely.

Remark. The proof actually implies that if Ω( 1n) ≤ p ≤ 1−O( 1

n), then the

probability that a graph distributed as G(n, p) admits a perfectly balanced

partition is exponentially close to 1.

In the next section we give a proof of the main theorem. We do not try

to optimize the constants throughout this chapter. Also, we omit floor and

ceiling signs whenever these are not crucial.

8.2 Proofs

For two disjoint sets of vertices S1, S2 ⊆ V (G), denote by e(S1) the number

of edges in the induced subgraph G[S1] and by e(S1, S2) the number of edges

of G with one endpoint in S1 and the other in S2. We start by noting that

given a partition (S1, S2) we have

e(S1) =∑v∈S1

d(v)− e(S1, S2),

and similarly:

e(S2) =∑v∈S2

d(v)− e(S1, S2).

Therefore, in order to have a perfectly balanced partition we need to

partition the vertices to two equally sized sets such that the sum of their

degrees is exactly the same.

A variant of the following simple claim is proved in [50].

Claim 8.3. If a1 ≥ . . . ≥ a2k, then there exists an equipartition [2k] = T1∪T2

such that |∑

i∈T1 ai −∑

i∈T2 ai| ≤ a1 − a2k.


Proof. For a subset R ⊆ [2k], denote s(R) =∑

i∈R ai. Consider an equipar-

tition (T1, T2) of [2k] minimizing |s(T1)− s(T2)|, assume w.l.o.g that s(T1) ≥s(T2). If s(T1) = s(T2) we are done. Otherwise find elements ai ∈ T1 and

aj ∈ T2 such that ai > aj, and define a new equipartition (T ′1, T′2) by swap-

ping them we get T ′1 = T1−ai+aj; T′2 = T2−aj +ai. Clearly, s(T ′1)−s(T ′2) =

s(T1)− s(T2)− 2(ai − aj) < s(T1)− s(T2). Due to the optimality of (T1, T2),

we get s(T ′1) − s(T ′2) < 0 and therefore s(T ′1) − s(T ′2) ≤ −(s(T1) − s(T2)).

This implies that s(T1)− s(T2) ≤ ai − aj ≤ a1 − a2k.

Given a graph G, a degree matching is a set M of disjoint pairs of vertices,

such that the degrees within every pair differ by exactly 1. The following

claim gives a sufficient condition for a graph to have a perfectly balanced

partition in terms of the size of a largest degree matching.

Claim 8.4. Suppose that a graph G contains a degree matching of size at

least ∆(G)− δ(G). Then G admits a perfectly balanced partition.

Proof. Given such a degree matching M , we can partition the vertices in

V −⋃M according to Claim 8.3, and get an equipartition such that the

difference between the sums of degrees in each part is bounded by ∆(G) −δ(G). Now we add the vertices from M to the partition, pair by pair, where

for each pair we add the vertex with higher degree to the set of vertices with

smaller total degree. Since the initial difference is at most the number of

pairs and in every round we change the difference by exactly 1, after the last

stage the difference is at most 1. The sum of the degrees of all the vertices

in the graph is even, hence we conclude that the two sets have exactly the

same number of edges, and the claim follows.

Therefore, our main result will follow from the next lemma.

Lemma 8.5. For every ε > 0 there are α(ε) > 0 and c(ε) > 0 such that for

every graph H on n vertices that satisfies ∆(H) − δ(H) ≤ αn and εn≤ p ≤

1 − εn

, a graph distributed according to G(H, p) has the following properties

with probability at least 1− e−cn:

• ∆(G)− δ(G) ≤ 2αn.

• G has a degree matching of size at least 2αn.

8.2 Proofs 133

In the course of the proof of Lemma 8.5, we need the following variation of

the standard bounds of binomial variables (see, e.g., [42] Chapter 1, Theorem

6).

Claim 8.6. Let X ∼ Bin(k, p) where n/4 ≤ k ≤ 3n/4 is large enough and

p ≤ 1/2.

• There is an absolute constant ζ > 0 such that if p ≥ 100n

then for every

z ∈ [kp− 2√np, kp+ 2

√np], Pr [X = z] ≥ ζ√

np.

• For every constants ε > 0 and B > 0 there is a constant ξ(ε, B) > 0

such that if εn≤ p ≤ 100

nthen for every 0 ≤ z ≤ B, Pr [X = z] ≥

ξ(ε, B).

Proof of Lemma 8.5. We will prove this theorem for p ≤ 12; the case p > 1

2

follows by taking the complement. The first item follows using Chernoff-type

bounds; For any pair of vertices u and v, we have |dH(u)− dH(v)| ≤ αn, so

for every p, the probability that the difference after modifying edges is at

least 2αn is exponentially small. Using the union bound we conclude that

the first property holds with probability exponentially close to 1, as claimed.

For the second item, let V1 = v : dH(v) ≤ n/2 and let V2 = V − V1.

We can assume that |V1| ≥ n/2, the other case can be treated similarly. Let

U1 ⊆ V1 be an arbitrarily chosen set of size n/4. Clearly, for every u ∈ U1,

there are at least n/4 vertices outside U1 which are non-adjacent to u.

We expose the neighbors of U1 in G in three stages. At the first stage,

we expose the edges inside U1. For every vertex u ∈ H, denote by Nu the

set of vertices outside U1 that are adjacent to u in H, and by Su the set of

non-adjacent vertices to u outside U1. At the second stage, for every u ∈ U1

and v ∈ Nu, we delete the edge uv independently with probability p. We

denote the degree of u after this stage by d∗(u). Let b = max 10, d√npe.We group the vertices of U1 to bins of width b, where the bin Bi contains all

the vertices that satisfy d∗(u)+ |Su|p ∈ [bi, b(i+ 1)). That is, the vertices are

grouped according to their expected final degree. We say that Bi is heavy

if |Bi| ≥ b8. Recalling that |U | = n/4 we deduce that there are at least n

8

vertices inside heavy bins.


At the third stage, for every u ∈ U1 and w ∈ Su, we add the edge uw

independently with probability p. The degree of u after this stage (which

is also the degree of u in G) is a random variable Xu = d∗(u) + Yu, where

Yu ∼ Bin(|Su|, p) and all the variables Yu are independent.

A set M of disjoint pairs of random variables is a matching if every two

variables within a pair differ by exactly 1. We next claim that the set Xucontains a matching of size at least 2αn with probability at least 1− e−c(ε)n.

This will finish the proof as it proves that there is a large degree matching

of the vertices of U1 (and therefore, of the vertices of V ).

We first prove that the expected number of matched variables is linear

in n. Let bj, bj + 1, ..., b(j + 2) − 1 be an interval that corresponds to a

heavy bin Bj, and for bj ≤ k < b(j + 2) denote by Tk the number of vertices

from Bj with degree exactly k after the third stage. A degree matching can

be constructed by matching vertices of degree bj + 2r to vertices of degree

bj + 2r + 1, and therefore by linearity of expectation the expected size of a

largest matching in Bj is at least

b−1∑r=0

E[min Tbj+2r, Tbj+2r+1]. (8.1)

We note that the intervals that correspond to the bins are not disjoint,

though the bins are; we define it so to avoid a problem where most of vertices

in some bin Bj have degree bj + b − 1 and p is very small. Let Aj be|Bj |ζb

, if p ≥ 100n

, and |Bj| · ξ(ε, 100) otherwise, where ζ and ξ are defined in

Claim 8.6. In both cases it is easy to verify that Aj = Θ(|Bj |b

). Since the

final degree of every vertex u in Bj is distributed as d∗(u) +Bin(|Su|, p) and

n/4 ≤ |Su| ≤ 3n/4, the expected value of each Tbj+r, r ∈ 0, . . . , 2b − 1, is

at least Aj by Claim 8.6. We consider the following two cases.

Aj ≤ 16. In this case we have |Bj| = Θ(b). The probability that a certain

vertex v will be of a certain degree in the corresponding interval is Θ(1b),

where this probability may depend either on the constants ζ or ξ. Also, the

degrees of every two vertices from Bj are independent. It is fairly easy to

show that with some positive constant probability (that depends on either ξ

8.2 Proofs 135

or ζ) both Tbj+2r and Tbj+2r+1 are non-zero. Therefore, the expected size of

a largest matching is linear in b (and hence in |Bj|), as desired.

Aj > 16. For a fixed k, as E[Tk] ≥ Aj we have

E[Tk]− 2√E[Tk] ≥ E[Tk]/2 ≥ Aj/2.

Note that since Tk is a sum of independent indicator variables, the standard

deviation of Tk is bounded by√E[Tk]. Therefore, by Chebyshev’s Inequality,

for every 0 ≤ r < b, with probability at least 3/4 we have Tbj+r ≥ Aj/2. We

conclude that with probability at least 1/2, a pair of adjacent values Tbj+2r

and Tbj+2r+1 are both at least Aj/2, and therefore by (8.1) the expected value

of the largest matching in Bj is Ω(|Bj|).

To summarize, we proved that the expected size of a largest matching in

every heavy bin Bj is Ω(|Bj|). Denote by Z the maximum size of a degree

matching in U1. Since there are at least n/8 vertices in heavy bins and the

expected value of Z is at least the sum of expectations over all heavy bins,

we conclude that E[Z] ≥ c1(ε)n, where c1(ε) is a constant that may depend

on ε.

Next we will prove that the size of the largest matching is concentrated

near the expectation. We use a vertex exposure martingale, where Zi is the

expected size of the largest degree matching after having exposed the first

i vertices from U1 and their neighbors, and Z = ZU1 is the size of a largest

matching. Since exposing the edges of a certain vertex may change the

expected size of a maximum degree matching by at most 1, the martingale

satisfies the Lipschitz condition (see, e.g., [13], Chapter 7). By Azuma’s

Inequality, for α = c1(ε)/4 and c(ε) = 8 · c21(ε), the size of the matching is at

least 2αn with probability 1− e−c(ε)n, and the lemma follows.

Proof of Corollary 8.2. Again we consider only p ≤ 12, otherwise the

proof will follow by taking the complement. The case p = Ω( 1n) follows easily

by taking H to be the empty graph. For smaller values of p, every graph

distributed as G(n, p) asymptotically almost surely has n − o(n) isolated

vertices and is bipartite (see, e.g., [74]). Assuming that the graph has these


properties, we have two sets W1,W2 of vertices that are independent and

contain all non-isolated vertices of G. Next we add the isolated vertices

to W1 and W2 and get an equipartition (V1, V2), such that V1 and V2 are

independent and thus the partition is perfectly balanced, as desired.

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